Firstly it is nothing to do with "remote administration mode", and all that, so don't worry about that! It seems to have changed a little since 2000, indeed as you say, you don't even get that option in TS setup for 2003.
Secondly, I am a relative new comer to all this so, please don't blame me if my advice is way off!
Anyway, I used Citrix Connection Configuration to do this...
I think basically, this version of Citrix on 2003 "hijacks" the TS setup, so you kind of have to allow TS through Citrix...? That make any sense to anyone else, or have I got that way off the mark?
I take it you have given the relevant user/group permission to log on to the application/desktop you are sharing? In Citrix Management Console? (right click on the application, go to Properties, then Users...?).
Also, I presume you have TS licences and aren't just using the built in one's with the grace period? These can only be used by Administrators (possibly even just Domain Admins, can't remember).
Just so you know if your problem is the same as mine, I couldn't connect via Terminal Services either, until I allowed this option in Citrix Connection Config, if doing this then enables you to connect via TS, chances are it is the same problem, if you do this and still can't connect using anything but Domain Admin level accounts however, I'd double check your permissions/licenses.
Let me know how it goes!
Cheers,
Kris.