courage vs. courgeous

Yesterday at institute the teacher posed an interesting question: what is courage.
What came to my mind was: pushing forward and doing something despite fear and we started talking about fear and if its presence was necessary to have courage. I mean, if a person does something and has no fear while doing it-are they exhibiting courage? One might say that even if there is no immediate fear felt, as long as there is a possible negative outcome in doing something and you do it anyway it's a type of courage. I dunno...I would say that there needs to be a presence of fear-doesn't matter about the possible outcomes-if you're afraid (even if irrationally) and you go through with it-ur showing courage. Fear (anxiety/worry/stress etc.) is the key (in my mind).

Example: if you're not afraid of spiders and you see one and you smash it-you wouldn't think: wow i showed a lot of courage back there! you wouldn't even think of it-could the spider have somehow evaded you and bit you etc....well yea, the possibility of it was still there but it didn't worry you. But for me, it still takes courage for me to kill a spider.
That being said, is the definition of courage the same as "courageous"?
sorry if this is hard to follow-just letting my thoughts run here.

here's what came to mind: (based on my def. of courage above) it is totally possible to not exhibit courage and still do something courageous and it is possible to exhibit courage and do something the opposite of courageous.

Example A.) Hypothetically of course, let's say that someone runs into a burning building and saves a child. It's totally possible that this person did not exhibit courage- if they had no fear whatsoever, right? (forget how unlikely it is) like say...somehow they knew that they were not going to die/had no fear etc of the situation. Sure, the rest of us would assume wow-they have so much courage but we only assume that cause we know we would be afraid in that situaton ourselves. BUT this person wasn't afraid-didn't need courage. But this person DID do something courageous.
Example B.) Let's say some young punk gets into some gang and it's his first kill. He's scared/nervous-of reprecussions, whatever but he pulls himself together and does it. Now as much as I hate to say that he has courage-if ur going with the doing something despite fear definition then technically he showed courage. But was it courageous??? well of course all of us are yelling: no! why?
because he did something immoral. In order for an action to be labled "courageous" it has to be morally right- fear is not a requirement since we would all agree that that person who saved a child did something courageous but they weren't afraid/wasn't hard for them to do.