On the wikipedia page it says :
Nevertheless, it may have influenced the behaviour of later naval officers by helping inculcate "a culture of aggressive determination which set British officers apart from their foreign contemporaries, and which in time gave them a steadily mounting psychological ascendancy".[15] This in turn may have contributed to the success of the Royal Navy and the acquisition and defence of the British Empire, as commanders knew that while there was a chance of failure in battle, not to risk battle was certain to result in punishment.
How true is this? Also I've heard that due to this even small ships would attack ones that far outclassed it, are there any examples of this? Also if this was the case if a small ship found an enemy fleet would they be expected to fight or report the location of the fleet or something like that?
Byng's execution had a profound effect on the Navy of the mid-1700s. For one thing, it taught admirals (and other officers) that no matter how good their political friends, they could not escape the consequences of failing to act, and could not be perceived to be shy of offering battle. If there were plenty of things that could go wrong when attacking the enemy, on the other hand it would be a fatal error not to attack. The culture of determination in the face of adversity that the British navy cultivated over time certainly owes something to the example of Byng, though how much it owes to that is hard to quantify; it was reinforced throughout the decades by continued success in the face of even superior forces. Over time, the British assumed that they would attack whenever they saw the enemy, while the French and Spanish (and other foreign fleets) expected to be attacked and more than half expected to lose.
Also I've heard that due to this even small ships would attack ones that far outclassed it, are there any examples of this?
In general, ships that were of the same class were expected to fight one another. There was an unwritten rule that ships of the line, for example, would not engage frigates or smaller ships in a fleet action (which leads to the famous "Sink that brute, what does he here?" comment from the battle of the Nile, when a French frigate fired upon a British battleship). However, in some cases smaller ships in combination would attempt to attack a larger ship. The most famous case of this is when two British frigates under the command of the senior captain, Edward Pellew, harassed the French 74-gun-ship Droits de l'Homme and eventually drove it aground in a storm.
Also if this was the case if a small ship found an enemy fleet would they be expected to fight or report the location of the fleet or something like that?
Smaller ships, such as frigates, were often employed as scouts for the main battle fleet. If a frigate employed on a scouting mission sighted the enemy fleet, it had two jobs: first, to relay the sighting to its admiral and the main fleet; second, to keep in touch with the enemy fleet and continue to pass back scouting reports. It would not be expected to fight a fleet on its own.