i got the normal version of diablo 4 on battle net?!

I think they need to be on the same platform. Someone just asked the same question on Adam Fletcher Twitter - go read some of his recent tweet comments.

Not too sure but i think you’ll probably have to buy the game again if run on different platform. If i want to switch from pc to ps5, i have to buy the game again. What an amazing company.

do you need the steam version because you are playing on steamdeck?
you can also install battlenet launcher on steam deck and play the battlenet version on steam deck
there are many youtube tutorials out there.