I've been confused with this for a while so here goes...

How does playing different chords under different scales affect what scale you are using?

In a separate thread I saw that if you play A minor over an F chord, you are playing in F
Lydian. How can you tell? What makes it F Lydian? I didn't want to hijack the thread so I just made this one.

Yeah. You have to think of it as what would happen in sheet music. Sheet music doesn't care what the guitar fingering is like.

That's one disadvantage of the guitar. There are all kinds of ways to play a scale, but guitarists often consider a certain position a certain scale, when in reality, that position could be many scales, depending on the phrasing and/or chord underneath.