If you go to:

http://<your website>/please-log-in/

and then click on "Login / Register" in the top-right menu, you will see that the popup window doesn't look right.

I know that the series of steps don't make sense, but I don't think that it should work that way.

I see what he means. Screenshot attached. I came across this before, but I didn't think that people would get to it by doing those steps, like you said. But if it's going to show up, I agree with you that it should have the info in there.

Here's what I see on yours. ( I'm using Safari)

I always get a full page on my site when I need to log in:

I see now the login / register link leads to this.
I never understood why we needed both the Please Log in  and Login / Register links
 HMMM

That's always baffled me too if I'm logged in shouldn't the please log in disappear?

Unless that's another WP thing I don't know or does that only happen with the SY theme?

If I am logged in, I see "log out" where login/register would be if I was not logged in.

Is it something you can remove in your above header menu ?

If I am logged in, I see "log out" where login/register would be if I was not logged in.

Is it something you can remove in your above header menu ?

Maybe I am misunderstanding, but why would you want to remove the ability to log out?

Somehow the login popup became a victim of my customization. Visitors will be directed to the full login page instead which might be annoying when the person just found an image he wanted to buy.

Can anybody give me pointers how to reactivate/re-install the popup login window?

I think if you deactivate the theme, and re-activate, it reinstalls those pages. But make sure you have any custom code backed up first, or are using a child theme (and I would back up that code too). It's worth a try, but I can't guarantee it will work. There have been many changes since I had that issue. And believe me, I lost a lost a lot of things and had to keep reinstalling and reactivating before I was done.