Which cookie should I keep to prevent new logins at every browser start?

I see a section in Cookie for "1password.com", which includes several .accounts.1password.com cookies (I have multiple accounts, so you may only have one item there). You are correct, the 1Password.community cookies are only for this forum. Have you signed in since the last time the cookies were deleted?

I also see the safari web extension named with a UUID like you mentioned. I opened that database in an inspector and it appears to be the local storage associated with the 1Password Safari Extension. The UUID is likely auto generated, so it may change if you ever reinstall the extension.

Unfortunately I don't see any 1password.com cookies for Safari...

(I do see them for 1password.community, but those are only for this forum?)

There's a web extension with a UUID though; can that be 1Password related?

safari-web-extension_<UUID>_0.localstorage

Have you tried allowing the 1password.com cooke/cache?

This:: https://1password.community/discussion/123854/new-1password-sign-in-from-safari-email-every-time-i-start-safari

Turns out it’s not just a mail, but also a “new” (Safari) “device” on 1Password.com after every browser (re)start.

Can you describe the bug you are seeing in more detail?

It would help so that we can test and determine the best path toward fixing it.