Wow... I'm not blaming you, but people giving you copies of software like that is usually a violation of licensing agreements — it definitely would be with my templates (even the ones that are free). (The responsibility lies with the licensees.) In this period of severe inflation, people like me are struggling to stay afloat... just sayin'...
I'm fairly sure FxFactory plugins need to be installed through FxFactory. There may be some that are free that bypass the FxFactory copy protection code (watermarks and such). Other similar would be MotionVFX and Pixel Film Studios — installers are involved.
For all others, can devise your own categories if you want. Categories are simply folders created inside each template class folder inside the Motion Templates folder.
That said, if you're working with your clients original projects as well, then in order to maintain functionality with templates already applied, then you will need to have the *exact same name* categories that exist on your clients' systems.
You can have the exact same template in two different categories and FCPX will see them as two completely separate templates. If one used in some places is suddenly "missing", then you'll see the red warning overlay for that version where all the others will continue to be normal. Furthermore, parameters for missing templates will not be displayed in the Inspector.
What you are suggesting is not really practical. Not only will you have to live with the quagmire (if you want to keep all parameter settings as they were), or you'll have to organize the templates how you want them, leaving all the others in place, reapplying them to the storyline, readjusting all the parameters to the new location templates, then deleting all the older templates. At which point you can delete them from their installation locations and restart FCPX (and hope you didn't miss anything — especially if templates are being used in more than one project.) You should also be very careful about which versions you do use for your individual clients — if they don't have your setup, then they'll end up with the red warnings for missing templates.