Ok, by playing around with cookies, I can be simultaneously logged in on
https://www.frozenincarbonite.org and logged out on
https://frozenincarbonite.org. ... So it may be it's some combination of browser settings that treats them differently. For the moment, I've changed the home page link to have the www part (
@Horton). Please let me know if this fixes the issue.
(However, looking around, I can find lots of people complaining about logged out on Safari last year... you may want to look into your browser cookie settings, especially if this doesn't fix it.)