I haven't played the Ruy Lopez at all but I am curious about how it compares to the Italian. I've heard that one reason the Ruy is supposedly better is because White has an easier time playing d4 in one go, eg. in the Two Knights Defense White plays 4.d3 to protect the e-pawn whereas in the Morphy Defense White can simply castle and ignore the threat to his e-pawn. However, what if Black scares you away from playing d2-d4, eg. by threatening to enter the Berlin Defense / Marshall Attack? If White has to play d3 in those lines, doesn't he just end up in a position that's more or less equal to the Italian?
The big difference in my opinion between the Italian and the Ruy Lopez is that you're tempting black into weakening their queenside pawn structure with a6-b5. Attacking that extra hook with a4 is a huge source of play on the queenside for white that you do not get in the Italian after both sides for instance have played a4 and a5.
Yes you are correct though that another huge difference is the ability to play d4 in one move. However, since the Marshall and Berlin have entered the field, top players at least are content with just sticking with playing d3. So there's no huge difference between the Italian and Ruy Lopez there.