File permission management. Can't change files within WP

I’m trying to switch hosts. I moved files and database and all seems to be working fine, but when I login to WordPress and open a child theme file I get a warning like (translated from Dutch): You need to make this file writable before you can save it. See the Codex for more information.

I read the Changing File Permissions codex page, but can’t figure this out myself.

All my folders have 755 rights and files 644. The GUI and GID of all WP related files are set to my user account (not apache or anything). When I set a file to 666 I am able to change it from within WordPress, 664 doesn’t work, so it needs to be open to all users.

From the codex:
Any file that needs write access from WordPress should be owned or group-owned by the user account used by the WordPress (which may be different than the server account).
What is the user account used by (the?) WordPress? How do I set this? I don’t want any user to change my files, so 666’ing all my files isn’t a solution.

Thank you for your time!

(Asked this a week ago in the How to and Troubleshooting forums, but didn’t get a reply. This forum seems more suitable.)