Challenging the Term "Hero" in Beowulf

Like many a reader, I read Beowulf and was in awe of his might, his agility, and his lung capacity (seriously, the dude swam for a week). I also saw that while Beowulf is not of the land under attack, he seems to feel that it's his responsibility to protect it. On the surface, Beowulf is formidable, tough, and gritty. Beowulf no doubt inspired many other archetypal "heroic" saviors, but is he a hero? Merriam-Webster, the great and powerful, says that a hero is "a person who is admired for great or brave acts or fine qualities." Let's pick this apart, shall we?

So a hero must be brave, great, and fine, eh? Or at least their deeds must be. Beowulf's fighting Grendel can be seen as a brave act, since Grendel is a demon with a bloodstained track record. Grendel terrorizes the hall for years and years, and shows no signs of slowing down. Beowulf has undoubtedly heard the tales, but he arrives in Heorot with nothing but the most cringeworthy hubris.
Beowulf proclaims: "Now I mean to be a match for Grendel, settle the outcome in single combat" (29). When I read this, I cocked an eyebrow and muttered "single combat?" But it did not stop there.

"Hand to hand is how it will be, a life-and-death fight with the fiend"

.........What does Mr. Ye Olde Superman do? He throws his blade to the ground and takes Grendel on mano a mano! Greco-Roman style! YEAH!

........Does that not sound a little suspect? 

That, in fact, was the moment I understood that Beowulf was not completely like your garden variety superhuman hunk. If I were to save a town under attack from a monstrous, cannibalistic demon, wouldn't I take every chance to ensure my victory (and, moreover, the citizens' safety)? The tide and tone of character changes when Beowulf announces his plan to fight Grendel unarmed, and I began to see him as less of a hero and more of a brash collector of trophies.

I am reminded of Inigo Montoya in The Princess Bride. Inigo (for those of you who have not seen this masterful work of cinematic art) is a brilliant swordsman from Spain whose fencing skills put him among the elite of the world. While he is vengeful and headstrong about finding his father's killer, his character contains a Beowulf-esque element of hubris. When Wesley (then in disguise) finds Inigo, they engage in a dazzling sword fight. But before this takes place, Inigo mentions to his fellow cronies that he will duel Wesley with his non-dominant hand. I believe his exact line was:

"I'm going to do him left-handed...It's the only way i can be satisfied...If I use my right...over too quickly"

Now why would Inigo make that decision? Was he heroic for doing so? 

Why would he make the decision to put himself at an apparent disadvantage for any other reason than pride? Inigo and Beowulf share the desire to be known, feared, and respected. They would save a town so the next town could know about it. What makes them different is Inigo grows into a more 3-dimensional character after defeating the man who murdered his father, while Beowulf just fights a dragon with the same attitude he had 30+ years prior and gets killed in the process. 

It's difficult for me to think of Beowulf as a hero now, but I'm curious to hear everyone's thoughts. Are his acts more heroic in the connotative sense of the word? Does saving someone, regardless of your intention, make you a hero?