No, that's not what happened.
Due to EL rules, if a game starts, it must be finished.
So, the game started on Tuesday night, got a half inning finished and was rained out for the rest of the night.
The next day (starting with the bottom of the first) Wednesday's starters took over as if they were starting the game.
He was pitching on normal rest.
Okay! This article confused me when it said "Named an EL All-Star earlier in the day, Peacock took over for Erik Davis following a 45-min rain delay in the middle of the first inning. He proceeded to pitch four hitless frames before Wes Hodges led off the fifth with a single."