Not completely sure what exactly you mean by “different”, but no, they shouldn’t be different in general.
Hm…
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Just to be sure: Are you indeed talking about logging in and logging out regarding your browser extension? Or do you mean locking and unlocking with your PIN - and not with your master password? (locking and logging out - and unlocking and logging in - are not the same… and login only works with your master password and not with the PIN)
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Are you sure, you wrote the correct email address that is associated with your Bitwarden account? And that you wrote your email address correctly (no typos)?
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Did you try it with a different browser?
If those things don’t help - here is a more comprehensive list of things you could try: [Guide] "I can't login" - Some tips for login problems/issues