This is an interesting question! The quick response is that no, this isn't necessarily the way negotiations were conducted, but negotiations, and generally any intercourse between officers was highly formalized and subject to rules and expectations of gentlemanly conduct. It'll take some unpacking.

First, we have to remember that officers in most state armies were gentlemen of a particular class with a particular upbringing that taught a lot of unspoken rules about personal conduct; second, we can see formalization even in the correspondence between officers in the same armies; and third, negotiations between enemy armies, especially when formally requesting parleys or demanding surrenders, also followed a fairly predictable formula.

War is a Gentleman's Vocation

One of the more difficult things to understand about warfare in the long 18th century (roughly 1700-1815) is how polite it is. It's a strange juxtaposition, because on the one hand officers are blowing each other to atoms with cannons and trying as hard as they can to bayonet each other in their sleep, and on the other, each one genuinely acts like their opponent is a peer on a ball floor, not someone trying to kill them. Modern culture, in many respects, simply lacks this level of peer interaction. Interpersonal relations are far more relaxed, and while there are still unspoken rules for interaction that have their basis in a shared (perceived) peerage, it's not as rigid as it was in, say, the 1770s.

This was a part of masculine public intercourse in every context, not just in warfare. You are polite because that's how a gentleman acts. Gentlemen aren't just rich people, or landed people, they are members of a social group with burdens of responsibility that extend from private to public spheres and vice versa. It's one reason (of many) that the British military relied on gentleman volunteers to serve as officers who had to purchase their commissions. It was a deterrent for those who would use the military for purely social-climbing purposes, establish that the people applying were actually, genuinely interested, and it would also take some of the financial burden off the state when it needed to raise men for some emergency.

All of this meant that men of the officer class were already thoroughly schooled in masculine intercourse before ever they wore a uniform. It had been taught to them as boys and as young men, bolstered by a life lived unavoidably in the eyes of a peerage that would punish any mistake in a variety of ways ranging from subtle (a snub at a ball, for instance) to extremely unsubtle (a challenge to a duel, for instance).

As a result of all of this, a sort of panopticon of peers constantly judging you, the risk of shame being unbearable and ruinous - because your public conduct was connected to your private life, your income, and future prospects, and those of your entire family - formalized ritual was sort of a reliable fallback. You don't have to like anyone, but you have to respond to an invite to their ball in a prescribed way, you have to greet them in a formalized way, and be thoroughly polite in all your interactions. This, in a weird way, makes things easier! You don't have to constantly judge the context, it's all there, provided for you, you just follow the (unwritten) book of etiquette.

And so, when these same men are in scarlet uniform coats staring down the barrels of the enemy's gun line and demanding that they give up a hopeless fight, you use the same rules, because they're not rules of war, they're rules of gentlemanly conduct, which is an international peerage based in ideas of social purpose that date back hundreds of years. Europe - and to a lesser extent, the American colonies of the British and French and Spanish - still ha a great many ideas about the organization of society, and though the Enlightenment was changing many of them, one of the harder to eradicate was the model of the three estates; basically God organized human culture with laborers on the bottom and the clergy on the top, and in between, the second estate, were the gentlemanly class. They used to be knights. It was still believed to be their unique social burden to fight wars, and given the rigid social context of most young gentleman's upbringing, the idea that they were meant to fight wars meant, in many cases, an early and enthusiastic schooling in the various gentlemanly arts of hurting other men. Riding, shooting, fencing, were all staples of rich young men's education. Dancing was a way to learn formalized social intercourse with young ladies, a must in polite society, but also because it taught you your own body, which made you a better fencer, it helped form your limbs with strength and dexterity, it schooled you in rote learning and discipline. On top of that, you learned mathematics, learned how to manage a house or a business, learned navigation, maybe, or trigonometry, studied textbooks on warfare and artillery and read history, and learned other languages to better study their culture's outlook on war. This was a time of great belief in what eventually became race theory; learning how the French fought wars was learning the unique cultural characteristics of the French, their essential racial temperament, which could give great insight on how they conducted a war.

The result of all this was an entire culture of men who believed that God made them in order to fight wars, and were raised with that expectation, and were educated with that expectation, and so when they finally did it, it was conducted with an odd (to us) formality that was a learned response to remove emotion and bias from stressful situations. A gaffe at the ball might not mean you get ripped open by grapeshot, but it might mean your marriage prospects do; war is in some ways less pressure than that!

My compliments, and would you go kill those men?

It should be no surprise, primed as we are now, that formality was as entrenched in international discourse as it was internal discourse. That is, conduct among and between officers of the same army was just as formal and ritual as sending compliments to enemy officers. Even orders are given with attention paid to polite discourse.

"I have the honor" marks the head of nearly every written correspondence between officers, even insofar as acknowledging letter: "I had the honor of receiving your letter of the 15th instant," for example, or signing letters with, "it is my honor to serve," or "I have the honor to be your obedient/humble servant." This is sometimes part of letters between brothers when they are serving together, such as Isaac Brock - the governor-general of Upper Canada at the outbreak of the War of 1812 - receiving letters from his younger brother James, his secretary. Letters are speckled with polite cageyness, talking about feelings and emotions relating to the situation, all adding a lot of extra (and perhaps, from a modern perspective, unnecessary and distracting) information to a report. For instance, here's a bit of a letter from General Henry Dearborn (he was American the officer in command of the north-east theater, essentially the nigara region to the lake champlain region) to General Stephen van Renssalaer, the commander of forces at Niagara, in the early phase of the War of 1812:

Sir, — Your letter of the 27th by Captain Dox, has been received. Before this reaches you. I trust the reinforcements will generally have arrived. I have made every effort in my power to have sufficient quantities of ammunition, medicine, arms, &c., forwarded, but not with all the success I could have wished. I shall continue to send on until I am satisfied there will be an ample supply. It rests with you to determine the proper time for acting offensively. You will, however, perceive the expediency of consulting the principal officers.

I am apprehensive that the enemy might attempt a stroke at the naval armament preparing at Sackett's Harbor, and it being of the first importance that no interruption should retard the progress of those operations, I doubt the expediency of withdrawing any part of the force from that place at present. When the troops destined for your post shall have arrived, your total force must exceed 7000 men, which I presume will be sufficient for all contemplated purposes.

I confidently calculate on a co-operation by way of Detroit, and on important aid from the naval department. It will be advisable to strike at Kingston from Sackett's Harbor or its vicinity, or at least to attract the attention of the enemy in that direction, by such movement as will threaten a blow. If we should be so fortunate as to obtain the command of the lake, Kingston and the country about it may be taken possession of and all supplies in that direction may be cut off. You should have as many flat bottomed boats (and scows if possible) as will be sufficient to transport 5,000 men with field pieces and artillery horses at once, with the aid of such other vessels as can be readily procured.

All emphases are mine, just highlighting some of the more evident points of humbling discourse. You can sort of see how cluttered things are. Orders aren't handed down by some supreme commander, they are given as advice between gentlemen of standing and intelligence. This is one professional giving professional advice to another; the fact that Dearborn is, by any objective measure, Van Rensselaer's superior, doesn't enter into it. Even orders given in the heat of battle resemble this kind of carefully phrased correspondence. It's not "charge those guns," it's "I humbly request that you position your men on that hill" or "general's compliments (to the receiver) sir, he wishes you to take position on the hill and silence the guns."

part 2, below!

This is a great question. I'll answer as best I can from the perspective of European and North American militaries during the eighteenth century (really between 1688-1789).

I've had to rewatch the scene in question for context, as a professional historian of the era, I feel like I need a bath.

Parleys were quite common during the course of eighteenth-century warfare, but they most frequently occurred in specific settings:

1. During the course of formalized siege warfare.
2. When one army needed to surrender to another.
3. When orchestrating the exchange of prisoners.
4. When two armies were camped in close proximity to one another, usually before a battle.

Depending on the circumstances, these communications could have a very formal or informal nature. I'll talk briefly about each of four scenarios I described in turn.

1) Siege communications. During the course of a siege, the besieging army and besieged garrison where in frequent communication. The most common means of asking for a Parlay was to send a drummer forward to beat chamade. A Military Dictionary, printed for Robinson in 1778, defines Chamade as follows:

"Chamade: a signal made by the enemy, either by beat of drum, or sound of trumpet, when they have any matter to propose. Otherwise called, to sound, or beat a parley, which is the more proper English, but Chamade begins to grow familiar, as do all other French terms in martial affairs."[1]

In the course of the siege, the governor of the town and commander of the army would be in frequent communication, haggling over the exact terms of surrender. This is the closest we come to the formalized ritual communication in the Patriot. The formalized issues which needed to be addressed were: Safeguarding the lives and property of civilians in the town, status of the garrison (free to leave or prisoners of war) property of the garrison (could they take it with them or was it to remain as loot) which gate would they leave by, were they allowed to carry loaded muskets and place a musket ball in their cheek (to show, honorably, that they had not been totally defeated) and finally, the all important question of what music should be played when during the capitulation ceremony. If the defenders were allowed to play a traditional march tune of the enemy army, it was a great honor.[2] This formalized negotiation was the peak of ritualization in eighteenth-century warfare. With that said, formalized nature took into account real power in the progression of the siege, if the besiegers had a difficult time reaching the glacis or breaching the walls, the defenders were more likely to have a favorable negotiation.

Asking a city to surrender too early was an embarrassing faux pas, as General Tattenbach reminded the Duke of Liria during the siege of Gaeta in 1734: "It is too early. You have formed no batteries and planted no cannon which will give me any reason to capitulate. I beg you, my dear general, to be patient and wait a little longer."[3] Since film plays a large role in your question, I would recommend watching this scene from The Last of the Mohicans. It isn't perfect, but you get a better sense of the negotiating objectives and tactics of each side in this type of communication than in the Patriot.

2) Army Capitulations. At several points in the eighteenth century, armies surrendered to their opponents in the field in mass. As an American, the famous examples which spring to mind are Saratoga and Yorktown, but it is just as important to remember Pirna and Maxen during the Seven Years War, or the Russian capitulation during the River Pruth campaign in 1711. During this process, army commanders would not actually meet with one another, but would carry on a frequent written communication with their opposite number. Often, the surrendering general would call a council of war. Burgoyne did this at Sararoga.

As in siege warfare, drummers and messengers would communicate between the armies, often blindfolded during their passage between the lines, to prevent the messengers from conducting reconnaissance. As in siege warfare, the defenders would try to obtain the most generous terms possible preferring to go free. This was indeed possible: Burgoyne's army, according to the original convention, was slated to return to Europe, and Tsar Peter "the Great" of Russia went free during the River Pruth campaign. Going into captivity as prisoners of war was equally possible, as happened (more or less) at Pirna, Maxen, and Yorktown. Once again, the amount of damage that the surrendering force to still inflict on the capturing army was the main driver in negotiations.

3) Exchange of Prisoners. In the normal course of European warfare, the quick exchange of prisoners defined military conflict. During the Seven Years War in east-Central Europe, and the American War of Independence (as Cole Jones has recently demonstrated), prisoner of war exchanges were sometimes disrupted by realpolitik or ideological concerns. Americans have long given Cornwallis flak for failing to come and surrender in person at Yorktown, there doesn't seem to be a similar awareness that the colonial government revoked the parole of an entire army (Burgoyne's) after the surrender negotiations had already taken place.In the scene from the Patriot that you reference above, Mel Gibson's character returns dogs to General Cornwallis. This is actually based on a historical event which occurred in 1777, when General William Howe's dog accidentally followed American troops retreating from the battle of Germantown. As opposed to a formal ritualized communication, or claiming of aggrieved status, George Washington simply sent the dog back to Howe under a flag of truce. You can read the note which was included in the return of the dog here.

Frederick II "the Great" of Prussia was less fortunate during the War of Austrian Succession, when the Austrian captured his whippet, Biche. Biche was finally returned to Frederick with the conclusion of the peace, but spent a long time in Austrian "captivity."[4] Thus, obviously, the formalized negotiation for the return of the dogs is overdrawn in the Patriot.During the Seven Years War in Europe, the Prince de Ligne commented on the exchange of prisoners: "it was something of a joke in the English and French armies to be taken prisoner... you had supper with Prince Ferdinand of Brunswick [the enemy commander] and the next day you were back with your regiment."[5] Most parleys which involved the exchange of prisoners were short, and again involved the flag of truce and beating of Chamade.

4) Proximity Meetings. Perhaps the most informal parleys of all were those which occurred when the enemy army was camped near your position. Here, it was not uncommon for soldiers to talk, and exchange luxury goods (such as tobacco) between the armies while on sentry duty. [6] Junior officers would approach their counterparts for brief discussions. I am not familiar with any army-level commanders approaching their opposite number for an in-person meeting in this context.

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[1] A Military Dictionary, (London: Robinson, 1778), 36.

[2] For more on all of this, see: Christopher Duffy, Fire and Stone, (Edison: Castle Books, 2006) 152.

[3] Quoted in Duffy, Fire and Stone, 152.

[4] For a popular treatment of this issue, see Caroline Tiger's General Howe's Dog, (New York: Chamberlin Brothers, 2005). For a more thorough treatment of prisoners of war during the Revolution, see: T. Cole Jones, Captives of Liberty, (Philadelphia: University of Pennsylvania Press, 2018).

[5] Quoted in Christopher Duffy, Military Experience in the Age of Reason, (New York: Athaneum, 1988) 266.

[6] For more on this, see Duffy, Military Experience in the Age of Reason, 163.

While I'm not aware of the specifics of 18th-century surrender protocols, it's worth noting that Benjamin Martin is presented as a veteran of the French-Indian war, having fought as a British citizen on the British side. To that extent, it seems more than plausible that he would be aware of the basic contours of military protocol, given that he was effectively a British soldier.

That said, I can point to an instance where the specific words and phrases used (claiming 'aggrieved status' to activate a new line of statements and claims in a formalized fashion) was incredibly important and even contested. In July 1776, General Howe and his Brother Lord Howe, respectively in command of the land and naval forces of the British at New York, decided to send an offer of truce to the Americans. The problem was that the letter was addressed to a "Mr. Washington," resulting in Joseph Reed (Washington's personal aide and secretary) refusing to accept it, as "We have no person in our army with that address," with Henry Knox adding that the British knew full well what rank General Washington held and expected to be addressed as. The British demurred and attempted to deliver a letter again, this time addressed to a "George Washington, Esq., etc., etc."; again it was denied. Finally, on July 20th the British sent Colonel James Patterson directly to meet with Washington. After the exchange of pleasantries, Patterson tried to give Washington another letter addressed to "George Washington, Esq., etc., etc." - Washington refused to touch it, claiming that a letter to him should be addressed to the station he held, not to him as an individual; the former was negotiation between military peers, the later private correspondence. While Patterson was dissapointed to be unable to deliver his message, he did report back to the brothers Howe that Washinton acted with "a great deal of marked attention and civility" in allowing his return, and was according to Henry Know "awe-struck" by the poise and dignity of Gen. Washington

I relate that story to make two points. First, as to whether highly ritualized communication between rival militaries occured: most definitely. The rules for late 18th-century warfare were complicated and involved a tremendous amount of deference to an opponent, as well as strictly codified rules for referring to military rank and title. Thus, even though simply calling Washington "General" would have allowed for negotiations between the British and Americans to begin, doing so for a non-comissioned leader of a provincial revolt was unthinkable, now matter the bearing of the man in question, the number of soldiers he led, or his record in the field. Even though Colonel Patterson expressed a wish to abandon an "adherence to forms" in order to allow "business of the greatest moment and concern," the professional military of the British demanded a standard of conduct that would permit no deviation from the proper "forms" of address and title. In other words, the ritual forms of negotiation and the symbolic overtones of the exchange were sometimes placed in higher regard than an actual diplomatic back and forth.

As to the question of how common such interactions were, it's harder to say. In the instances where ranking members of the American military or diplomatic corps interacted with their British counterparts, there was typically a tremendous amount of deference and respect to protocol on all sides, albeit with a fair amount of British complaining behind the scenes. The British officers learned this system in many cases from birth, as the rigid hierarchy of the military mirrored and intertwined the rigid aristocratic forms of British society writ large. Many of the Americans learned this system either from service in the British army (e.g. Washington), training from those who had been in service to the British Army (e.g. Reed learning from Washington), extensive personal reading (e.g. former bookseller Gen. Henry Knox), or prewar status as powerful British citizens. Regardless of where they learned to behave like proper military men, in the course of the conflict the Americans made a tremendous effort to not only win battles but to present themselves as a legitimate, organized, and above all respectable military force. In the context of the time, learning, understanding, and using proper military protocol and decorum were a critical, if not fundamental, part of becoming a respectable military.

Sources:

McCullough, David. (2005). 1776. Simon and Schuster Paperbacks. 144-146.