In the situation of Hector in Iliad, which would you choose—responsibility to people or responsibility to one's ability?

I would make the choice that Hector makes: responsibility to other people. This is one of the many personal qualities that makes Hector such a sympathetic character and which distinguishes him from Achilles.
Whereas Achilles is interested only in personal glory, Hector actually cares about the welfare of his men and of the entire Trojan people. And whereas Achilles sulks in his tent while his comrades-in-arms are getting slaughtered outside, Hector goes out of his way to avoid unnecessary suffering. If he needs to make a tactical retreat to save lives, then that's exactly what he'll do.
Hector also has to think of his family in all of this. He knows that if Troy should fall, then his family will be sold off into slavery. So when he's fighting bravely on the field of battle, he's not doing it for himself like the self-centered Achilles—he's doing it to protect his family and to prevent the Achaeans from turning them into slaves.