Just to be clear, say in the case of determination, I get dealt a face up pilot card, and it says discard it; that means it does not go face down as dmage, but goes away. Is that correct?

That is correct. The card goes into a discard pile instead of going onto your ship as damage.

Though it would not work in the event that a face down damage card was flipped face up, from say Rex's ability.

Discarded damage cards are cards you do not need to worry about as they go away.

There are also several effects that can turn face-up cards into face-down cards but there you still keep the cards.