The problem with the first usage is that it makes heroism something accidental. The hero is in every way ordinary, but then they are faced with sudden unanticipated hardship and they overcome it. Little attention is paid to what led to the heroic act. It was an event thrust upon the hero, and nothing prepared the hero for this heroism. They just chose well in a tight situation. When we use the word this way we undervalue the character of the "hero," and ignore their discipline and virtues (or lack thereof).
The second usage has three problems: First, it's obviously mistaken, as events like Abu Ghraib and My Lai should make plain. Second, as with the first usage, it diminishes the long, hard work of those men and women who live heroically through self-discipline and the cultivation of courage and moral character. Third, this usage is almost always cynical, and politically motivated. It is usually the politician who says it, and usually for the benefit of the politician. It is a shibboleth of political life to "honor the troops" or "salute the men and women in uniform" in words -- and usually in words only. Real honor and real salutes come through much harder means, like supporting them financially, emotionally, and spiritually.
Aristotle (correctly) said you can't really judge someone's life as happy until they have lived it all. It's probably similar with calling someone a hero. Certainly we should stop making statues of people while they're still alive. It should probably be a rule of life that we shouldn't call anyone a hero without serious public debate leading to consensus. When you compare the process by which the Catholic church determines whether to call someone a saint to the process by which an American politician decides to call someone a hero, it's pretty plain which of the two processes is more rigorous and which is cynical and thoughtless. At least the church engages in research first.
Maybe we should stop using the word altogether, because there is another problem that comes with most uses of the word: naming someone a hero practically divinizes that person, making it much harder for us to think critically about his flaws. This ought to concern all of us, and especially the one called a hero, who is thereby even further distanced from our common life.