Did the pay phone that the Facts of Life inherited from Coed Fever for the first 4 episodes before being taken down for the Nancy O./Roger hot line the same pay phone we saw in the cafeteria in Season 2 and beyond?

Short answer: No.

For a longer answer, see the detailed analysis with photos here in the Season 2 Props/Sets/Decor discussion thread - page 5, posts 69 and 70 (https://www.sitcomsonline.com/boards/showthread.php?p=5394403&highlight=phone#post5394403).

Hahaha!

'80s and I wondered this, ourselves. As he says, he discovered they are different: the receiver and the cable connecting it to the... eh, other part are not the same.