I checked and in the "shared" albums I had 3 albums and in "normal" I had all the remaining (approx 20).
I created just one more album as the "normal". The Ozio Gallery pop-up with option to make albums visible was still not showing.
I deleted the one album I created 2 days ago and recreated it, making sure it is "normal" and not "shared". After refreshing joomla, the pop-up eventually appeared. That's strange. I currently see that I still have few albums on Google Photos created as "Shared", but the gallery works normal.
For now the problem is resolved, but maybe it would be good to check why the pop-up was not showing at all (with all remaining 20+ albums which are "normal").
One more thing - the album I created now and made public on my joomla site is not set as publicly accessible on Google, but I can see it on my web page.
By the way - on the list of available albums in Ozio I see approx. 10 albums which are just dates (like 2015.07.14). When I look for such albums in Google Photos, I can't find them. Can you please write how Ozio gets the list of albums and how to eliminate these fake "date-albums"?