In TROS, the gang refer to the wreckage of the Second Death star on the Endor moon as simply ‘The Death Star’, presumably to not complicate things. I think that adds to the list of good reasons, not least of which being that under the ST in particular we have way too many Death Stars, that we should consider having the DS2 just be a part-repaired DS1.

In ANH, I’d edit the explosion to make it look more like massive damage than full destruction, and in ROTJ, I’d remove references to it being another DS, and add references to it being simply under repair.