You can give admin any permission you want. It's just that ulx commands with the "ULib.ACCESS_ADMIN" tag will be given to the admin group by default (which you can then remove if need be). You don't want to change the name of the admin group, that's all. I know there are sometimes issues with group names that have non-underscore symbols in them, so again that could be your issue.
Oh and also, putting "user" as a whitelist option is pointless, because "ulx adduser player user" is functionally equivalent to "ulx removeuser player".
Edit: And just so you know, you are right in the assumption that those values are what the group can add to. It's a whitelist, which is a list of available options, as opposed to a blacklist which is a list of unavailable options.