Short question, I can only run a server on my own PC and my own connection, is that correct?
Or can I also rent a server?
You can host a server from your machine, yes.
Just keep in mind if you want to run the game and the server at the same time, to always start the game first.
@Malarn to answer your question: yes you can rent a server to run a t6 server, you're not limited to your own pc.