You added the user correctly. I have no idea how else you would add someone to that local admin group.
The only other way to accomplish this would be to create an A/D group with local admin rights and add your user to this group. The net effect is the same, however, and as for assigning local admin rights on a system by system basis, I agree with MrDenny…
It depends on your intention. If you want people to be administators on any corporate Windows system that they log onto, then create a group in AD, add them to the group, create a GPO for your workstation OU to ensure that the group is a member of the local Administrators group. The policy for this has two modes, adding and overwriting. I suggest the mode to add the group.
This has the ssimplicity that you can pull an ID out of the group and you do not have to touch the systems.
Just ensure that you place all workstations in a common set of containers and apply the GPO.
This removes login scripts, direct touches to systems and other items from the equation.
If you want to get fancy, you can use WMI and group filters as well.