What exactly would be considered a discard effect when regarding the Ongoing plot twist Earth Enslaved. "Whenever an opponent discards a card, you gain 1 endurance."

Enemy, Mobolize and things of the like are obvious. Even Slaughter Swamp and so forth. But what about Arkham effects that force discards? Does it have to use the word discard?

Also, what about power-ups...from what I remember, discard a card from your hand with the same name, target character gets +1/+1 for this attack. That is how it is worded or how I remember it. If that is so, would that be considered a discard?

Yes. You're right on the money. Powering-up a character card is discarding. It is what the OCR refers to as a "game-based effect." They don't have to be "forced" discards, (a la FFoe plot twists) they can just be uses of Enemy/Mobilize/Slaughter Swamp/free recruits.

B.