When I demote a player to take someone off the DL, are they always immune to demotion penalties? Or does that only work if it's the same player who was promoted when the injury happened originally?

Has to be the same player you promoted. And you have to do it as the same time.

The situation you have to be most careful with is if you have 2 ML players go on the DL at the same time.  Say you promote AAA Player 1 when the first ML player goes on the DL and AAA Player 2 when the 2nd ML player goes on the DL.

When the first ML player is ready to come off the DL, it often makes sense you'd want to demote AAA Player 2, because he's the lesser player, which is why you promoted him 2nd.

If you break the pairing you established when you put someone on the DL & promoted, both AAA Player 1 & AAA Player 2 might get a demotion penalty when you send them down.

Makes no sense to me, but that's how it works.