My issue is that I have a worker setup on my account on Cloudflare. But I want to use a domain which my account do not own. But I do have full domain administrator access and Domain DNS access.

Pretty sure you can only create a custom domain for an active zone you control.

So that means that the domain have to be registered under my account?

The domain doesn't have to be registered under your account, but there needs to be an active zone for the domain in your Cloudflare account

Well how do I get that? What I am reading in Cloudflare Docs is that zone is just a another terms for "domain"

Sorry, there are two parts one is having the domain in Cloudflare registrar which I thought you were referring to by registered. The other part is having the zone/domain active on Cloudflare, which involves changing the nameservers of your domain to point to your assigned ones from Cloudflare

Someone else is paying for the domain and they setup that domain on to their account but gave me admin and DNS persmission to change setting/configs. So I could specified that the domain that I want to use is shared.

Do you have permissions to edit workers on the account with the domain? If so then you can make the changes there. The worker has to be in under the same account as the zone/domain

no. But how can I get that permission/role

Need to have

Workers Admin

permission on the account. Have to ask whoever gave you your current permissions to increase them

Well I have domain adminstrator access. But I can't find the Workers Admin permission when trying to edit permission for my account

Yeah it is an account scope role as workers are account level

No need for that. Currently transfering the domain under my account :P