Which affiliation to use when publishing, when invited professor at second university

If you get support (financial and moral) from both places, you should list them both.

You can write:

University XXX (your permanent address)

Current address:

University YYY (your current address)

If it happens that the paper is going to be published after you are back at your home university, then you can remove the current address (when you receive proofs for final corrections) and leave only the address of the home university. However, you should put in the acknowledgements something like that: (This is an example of an actual acknowledgement from a published paper)

Final version was prepared when the last author was staying at Mathematisches Institut der Universitat Bonn; he gratefully acknowledges the generosity of Humboldt Foundation and hospitality of his colleagues from Bonn.

I think you should list the institutions that supported any substantial part of the research presented in your paper. As for your other affiliations, I think you may want to list some or all or none of them.