Apologies off the bat for what surely constitutes a double post, but having posted in the specific Login thread under Addons (which is right at the bottom of the root thread list, below the international section for some reason) I'm not sure anyone has any reason to dig that deep and actually notice my question. So here's a link:
https://forums.modx.com/thread/97751/weird-login-status-inconsistency-temporarily-solved-by-clearing-the-site-cache
It seems the problem has sorted itself out, but all I did was update the time zone on the server. But I have a second site project that works just fine, always did, and doesn't have the correct time zone, so I presume it hasn't solved the problem, and therefore I fear the problem will reappear and screw me over.
Any helpful hints would be gratefully received.