You can not move clients to another group. Never edit anything in the user section, just add clients and resellers, ispconfig will configure the needed users automatically.
Also, I am a bit confused about the terms Client and User. Is it true that when one creates a Client, a User is also created?
Yes. As clients and resellers can not login to anything. They are just holders for limits and the postal address. Only the users(s) assigned to a client or reseller can login. So as I described above, do not change anything in the system user are manually as it is not neede to edit something there. Everything is managed by ISPConfig automatically when you create or change a client / reseller.