Note to self: the “Domain Admins” and “Enterprise Admins” groups aren’t the primary groups in a domain. The primary group is the “Administrators” group, present in the “Builtin” folder. The other two groups are members of this group and thus get rights over the domain. The “Enterprise Admins” group is also a member of the “Administrator” group in all other domains/ child-domains of that forest, hence its members get rights over those domains too.
So if you want to create a separate group in your domain and want to give its members domain admin rights over (say) a child domain, all you need to do is create the group (must be Global or Universal) an add this group to the “Administrators” group in the child domain. That’s it!