How can I stay logged in when I close browser window??!!

I’m not sure if this is a cookie issue or what, but every time I close my browser’s window I am logged out. I’ve checked the “remember me” button at least 100 times, but no difference.

This only happens on one of my sites:
https://www.forwardmotiontheater.org
But cookie/login behavior is normal for my other WordPress sites (I stay logged in even if I quit the browser — which is what I want).

Can it be my browser? Safari 1.3.2, osX3.9? I looked at my cookie list to try to delete the specific cookie, but didn’t find anything under my site’s name; what would the wordpress cookie be named?

Or some conflict/error in how the cookie is tested by wordpress?

I even tried the COOKIETIMEOUT plugin set to 1 year:
https://www.freemoby.com/2005/05/21/wordpress-plugin-cookie-timeout/
But the issue is the same. I am logged out the second I close the browser window. When opening the site in a 2nd browser window I am not logged in in the second window. Only tabbed browsing keeps me logged in, as long as it’s the same browser window.

Don’t even know where to begin….
Please help!