After more 'playing' I think there is a bug.
First note that in the following every time I open SoL I use a bookmark. Every time I open SF, Fine or Bookapy, I use the link on the SoL home page.
So, starting on the SoL home page. I use the link to go to SF, where I'm logged in. I log out. I close the page and open a new one, I go to Sol, I'm logged in. I go to SF, where I am logged in.
Same applies exactly if I do that with FS.
But.
If I do the exact same thing with Bookapy, when I log out, close the page, open SoL in a new page then go to Bookapy, I'm logged out and get the password error.
(Note: open new page = open new browser window)
So, whilst it might not be a bug, it seems Bookapy acts differently to both SF and FS.
Over to you Lazeez.