Hmmm interesting one: The point is ambiguous. That is, unclear of what you are asking. Let me explain: “I want you to record yourself while you are cooking” What do you want? The person to try to cook and record? Or a video of the cooking? I’d like to offer third, preferable sentence.. I want you to record yourself cooking This means you want the person to record themselves completing the action, I.e cooking. Make sense?

I agree that 2 is more grammatically correct and is how you should write it. For spoken English, either would work. In fact, for spoken English you could even say "I want you to record yourself cooking."

Assuming 1 is top, 2 is bottom, both are correct for spoken english but for written english, 2 would be correct. (If you say 1, a native speaker will not correct you because native speakers would say it both ways). For number 2, it is more correct because for the verb “cooking” the subject “you” needs to be specified

I would say they both are correct

The second one i can assure you