Why would you imagine that medieval armor wouldn't have made much of a difference? Armor was manufactured and purchased with a great degree of care and effort. A knight or man at arms capable of purchasing a suit of armor would try and obtain the best he could within his resources because it was literally a matter of life or death. A trained warrior clad in mail (or later, plate) was extremely difficult to kill. With the invention of plate armor, men at arms were able to abandon shields because their armor was so good at protecting them from arrows and other weapons. An enemy attacking an opponent in plate armor had no chance at all of hurting him if he was reckless. Manuals of arms from the late middle ages detail many different combat techniques to deal with a man in armor, ranging from grappling holds to tripping moves with polearms and specialized stabbing techniques for use with swords. The quote about knife fights is set in a modern context, where people stabbing each other will not (in 99.99% of cases) be wearing real armor. This was not the case on a medieval battlefield.

As for suvivability of kings on the battlefield, there are several factors you should take into account. First, kings would always have a personal corps of their own household troops that served as a bodyguard unit. These men were well-paid, well-equipped, and highly motivated. To let their king die or be captured would be a grave matter. When King Harold of England died at the Battle of Hastings (1066), his housecarls fought to the death instead of fleeing, even though the battle was clearly lost. On a pre-modern battlefield, without access to advanced communications technologies like radios and telephones, coordinating a major assault on an individual in the thick of a battle would have been a difficult proposition. Certainly, a king or other important noble could be identified from his personal banner, but it was another question entirely whether it was possible to reach that banner from some other part of the battlefield. At the Battle of Crecy in 1346, the French attempted to capture the young Prince Edward of Wales, but were prevented from doing so because of the ferocity of his retinue and a surge of archers charging forwards to protect their prince. At the Battle of Bosworth Field (1485), King Richard III attempted a desperate charge to kill Henry Tudor, but failed to fight his way through Tudor's bodyguard and died in the attempt. Most often, kings were only captured or killed on the battlefield when the bulk of their forces had been routed or slain, like when the Roman Emperor Valens died in the Battle of Adrianople (376).