What do "brave" (adj) and "courage" (n) really mean?

Courage is reckoned the greatest of all virtues; because, unless a man has that virtue, he has no security for preserving any other.

Samuel Johnson, quote provided by Vincent McNabb in the comments.

Courage is not simply one of the virtues, but the form of every virtue at the testing point.

C.S. Lewis, sourced on Wikiquote.

To my mind, at the testing point implies a test. There's no test if you don't feel fear. I like the M-W definition you give. Note, though, that the Cambridge dictionary says that bravery is defined by not showing fear. It says nothing about not experiencing fear.

Off the top of my head, I'll say that both a brave person and a courageous person may experience fear. The brave person won't show the fear; the courageous person might. The brave are, then, a subset of the courageous. Take that with a grain of salt: I just made it up, but it sounds roughly right to me.

Clearly there are many denotations of these terms, so any response is likely to be a subjective one, but my interpretation of the words is similar to what Mark Twain's in Pudd'nhead Wilson

"Courage is resistance to fear, mastery of fear - not absence of fear. Except a creature be part coward it is not a compliment to say it is brave."

I interpret that to mean that doing something when you have no fear doesn't make you brave. Depending on the situation it may make you foolish or foolhardy, but not brave. Your initial definition works for me in that sense.