We should probably also change "Non-admin action" to something else, for example "Anonymous action"?

Agreed. It currently looks like this in a Git commit:

Author: Non-admin action <nonadmin@example.com>

I'd probably:

1. Omit the word "action".
2. Find some good email address that would be stable and suitable for client-side UI rendering (we should compare email rather than name here).
3. Find a name. It's less important IMO than the email.

Something like:

Author: WordPress <vp+nouser@example.com>

("WordPress" is an idea after trying more obvious options like "Unknown", "Anonymous", "Non-user", "Unauthenticated" and similar; the word "WordPress" at least indicates that the commit was created by this CMS but I'm open to other options as well.)

I am not sure about naming. I liked "Anonymous" initially, but what about "VersionPress" (commit was created by VersionPress)? Email vp+nouser@example.com is fine, I guess (or vp+anonymous@example.com?).
I like the idea of comparing email (and not name) and removing "action".

I like the "VersionPress" suggestion, this is how it would look:

Author: VersionPress <vp+nouser@example.com>

What do you think?

BTW, are you comfortable enough trying to update the front-end as well? I think that the icon and tooltip are reasonable for now, just the detection should change to an email (and included the current nonadmin@example.com as well for compatibility).