Ps4 controller on Ps5?

I plan to get the Ps5 in the foreseeable future. Since I actually only play first-person shooters and go competitive there, I've only been playing with a scuf controller for about 2 years. So recently my scuf broke and I want to get a new one. But since I don't know when I'll get the Ps5, I wanted to get a scuf impact for the ps4 again for the first time. Now I have read that ps4 controllers are only compatible with the ps5 games that are also available on the ps4 and not for games that are only available for the ps5 (correct me if I'm wrong).

Now I've got the cross play bundle from call of duty cold war, with the intention of being able to play it on both consoles (that's logical.). In other words, because cold was also playable on the ps4, I should also be able to play with my Ps4 scuf on the Ps5 cold war, right?

I'm just not interested in ordering two controllers in the shortest possible time.

If you play PS4 games on the PS5, the PS4 controller can also be used. If you play a PS5 game with a PS4 controller, there may be restrictions.