I’m not using any type of cache plugin. I do have a redirection plugin that redirects from pages from our old design to our new design, but I only have two redirects and none of them are for the homepage, and I get the same behavior with this plugin deactivated. Fixed by adding home_url() to the wp_loginout function call.
Looking to do the same thing. Do you mind sharing what you did, so I can replicate it on my site? Or did you just follow the instructions in the link you provided?