In the English Opening, where should the d-pawn go? Whenever I play the English Opening, I usually try to achieve a c4, d4, e3 pawn structure. However, in this video by STL chess club, the instructor recommends playing d3 (and possibly e4 later) to control the white squares as much as possible.

Which is objectively better? Does it depend on the opponent's play?
Once you play d4 you are usually leaving the pure English opening and transposing to a queen-pawn opening.  Objectively there are positions where it is absolutely better to transpose to a queen-pawn opening but you can certainly build a repertoire around playing the pawn to d3 -- it may just be easier for Black to equalize in some lines.

The formation of c4, d3, e4 with the kings bishop fianchetto'd, knights on e2 and c3 was Botvinnik's baby and is a viable way to play.