Broad question but I will attempt to give a narrow answer: You are close but, at least in England, imagine separate packs of sight hounds staged across the landscape, hunters on horseback, and the final death caused by spear/sword/hunting knife.

Edward of Norwich "wrote" a book on hunting around 1413, which was mostly a translation of an older French book (Livre de Chasse, by Gaston III, Count of Foix, aka Gaston Phoebus), with several chapters added by Edward. It is the oldest English-language hunting treatise. His grandpa was King Edward III, and he died at the Battle of Agincourt. So his bonafides are pretty well established.

Among other jobs, Edward of Norwich held the title of Master of Game which would be sort of like the royal lord of game wardens, and some of the chapters in his book specifically dealt with the English style of deer hunting.

To summarize, royal (or high noble) deer hunting would be conduct in a forest; which is not analogous to anything in America. At one point, 1/4 to 1/3 of all of England was deemed royal forests, with exclusive hunting rights belonging to the royal family. And, a "forest" was not just pure trees or woodland, but instead could be a mix of trees, marsh, swamp, open fields, moors (heathland), etc.

Edward of Norwich described in Chapter 34, Chapter 36, and other parts of the treatise, basically as follows (with modernized language):

Hunt par force:

Before the hunt the Master of the Game will meet with various helpers. These could be other staff members, huntsmen, horse grooms, dogmen, etc. all the way down to any peasants who were giving assistance to the hunt day. He will gather information (where has anyone seen any deer lately, where are the biggest males, etc.). Someone (the Master of Game or his designee) will have gone about looking for tracks, and measured those tracks with a stick (presumably broken to the size of the track). He will also collect poop (described as "fumes."). He will keep the fumes in a sack or hunting horn, nestled in grasses he collects. He might even physically see particular deer in particular places (i.e., I saw the biggest stag down by the pond). Someone will alert the sheriff and other locals that a hunt is occurring, and someone will make sure a cart (for hauling the quarry), draft animals, and other logistical support is available.

There will also be a meeting between the Master of the Game and the Lord to explain the plan. The other members of the hunting party may be there. They will look at the sticks and poops, and decide which specific deer to hunt that day. After the huntsmen pick the specific deer they want (based on the size of the tracks, the diet shown in the poops, and other information), the Master of the Game can then use the information gleaned from the staff/peasants to stage fresh packs of sight hounds at various locations. He will then assign various staff to various locations where the chosen stag has been seen recently and where his tracks and fumes were found.

The hunt starts on horseback, with a pack of sight hounds. At least 12, preferably 24 of more. These would be greyhound-type dogs. Edward basically writes that the more hounds, the merrier music they make.

Another dog appears; a "lymer" which is kept on a lyme (a "line", aka a leash or rope). He is handled by a fewterer (a tracker). The fewterer is different from the man who handled the sight hounds. The lymer dog would have been a primitive or proto-scent hound; akin to a modern bloodhount or black & tan, etc. Once the lymer strikes the scent, a horn is blown and the sight hounds are released. They would be the proto-greyhounds (or Scottish Deerhounds, etc.). If they see the deer, they are loosed to chase him.

The chapter also describes lots of horn blowing. Horns to start the hunt. Horns to communicate across the fields. Horns to direct dog handlers to do various things with various packs of dogs, etc.

The hunters are now following the sight hounds, similar to an English fox hunt. If the game is lost, some horns are blown and the lymer shows back up to catch the scent again. Eventually, as the chosen stag crosses the forest, if he runs past a pack of fresh sight hounds, they are released. They are never released before he gets there. He eventually tires. Then someone stabs him in the heart; with a javelin/spear/sword/hunting knife. More horn playing happens. The dogs are fed some of the entrails. A return song is played on the horn.

Hunting/shooting in a deer park:

Now, if you are only a middling noble, you might never get invited to hunt with the king in the royal forest. But, you might be granted the right to keep deer in a deer park by the king; a grant of emparkment. This is a right to keep your own private herd of deer on your land, in your own little hunting preserve. Kings and high nobles also kept deer parks as well.

A deer park is an enclosed area where deer are kept for shooting; usually in sight or close to the main country house. Your deer park might be 100 acres of less, or might be thousands of acres and several miles across. (Note, even if the king's grant of emparkment allowed you to build a 500 acre deer park, often the actual parks were smaller). For reference, Central Park in New York City is 843 acres. Hyde Park, in London, is currently 350 acres. Originally, it was a deer park, was purchased by Henry VIII, and remained a royal deer park for private hunting for 100 years, until James I. During that era, Hyde Park was approximately 620 acres (for further reference, 1 square mile is 640 acres).

It would be fenced, and probably have ditching dug around the inside of the fencing (to make it harder for the deer to jump out). Some deerparks had ramps on the outside, allowing deer to jump in but not jump out. Some fences were also supported by thick hedges. Some estate owners stocked deer in their deer parks, while others only managed a "native" herd. Some used the parks for hunting in a nature preserve type setting, while others used their park basically as a pasture just to produce venison for the table.

In 1086, the Doomesday book recorded 36 private deer parks. Deerparks remainded in use until at least the 1500s. The Normans introduced the fallow deer to England, including stocking them in deer parks. Deer parks were used in both England and Ireland. Eventually they fell out of favor, probably because of economic pressure during/after the Civil Wars in the mid 1600s to make more profit from agricultural estates.

The deer park would be fenced, and the terrain and plantings often altered. I.e., many of the trees would be cleared, some brush or trees left in small clumps for cover, etc. The land might have been used as mixed pasture and crop land, or purely as a human-altered "nature" preserve. (Aside - this is not unlike modern-day Texas, where high-fence deer hunting on private ranches is common, and not unlike many modern American rural areas where food plots are specifically planted for deer and cover specifically provided).

When hunting in a deer park, the huntsmen would usually stand in a line with bows or crossbows; similar to a skirmish line. The text is unclear, but the hunters might be standing behind or inside of little brush piles (described as "tryste" or "standing"), similar to a modern blind, to hide them from the deer and provide some protection from the elements.

Your manor house/country estate was often inside the deer park. So, effectively, your mansion and its "yard" and surrounding grounds were inside a captive deer park. You could look out in your yard and sometimes see the deer (not unlike much of suburban America or people who live on golf courses. . .).

When the hunt/shoot started, the line could be convenient to the main house, so that spectators could watch and to make it more of a social event. The staff (with or without dogs) would start on the other side of the park, and march through, driving the deer to the line of hunters. The hunters would then shoot them on the run with bows or crossbows. This is similar-ish to modern-day pheasant shooting in the UK.

In the event that the King came to hunt in your deer park, you or your Master of Game was to ensure that the King had the best place to stand during the drive, and that you "ought to make fair lodges of green boughs" "to keep the King and Queen and ladies, and gentlemen's and also the (King's) greyhounds from the sun and bad weather."