In practice though, if using the self service machine, is there really a difference (other than the extra step of swiping the passport)? Do they actually have different cutoff times for the machines based on if it's domestic vs. international checkin?

That said, IIRC there are a few airports where the machines don't let you check in within 45 minutes of departure (not 30), and you have to then see an agent (who may not be willing to do it for you w/o a fight). I don't know what LGA does.