Thanks @igazor for the screenshots!
Update -
I found out what the problem was before - I did not have the "Move to" option because the household was more than 8 sims (note to self: don't travel with more than 8 sims). So I started all over from AP (the original hometown), travelled with 8 sims only, I did everything according to the instructions (managed to move them to the other vacation home, waited the animations to finish, then- clicked the city hall, change hometowns, save, and exit). After the saving- the icon in the left corner of the other town was not there anymore as before (only the one of University). They icons of all connected towns *should* appear in Edit town, right?
after I started the game all over (from the launcher) I tried one of the sims to travel to AP - he got to the empty town again... It is important to mention that after all this process- the relatives who live in the other town *do* appear in the relationship panel in grey, and in the family tree.
So maybe the problem (now) is with the CC world itself*???
btw I hope it's ok I'm writing all the details, it is important to me to figure which step exactly went wrong (maybe it will also help other simmers too...)
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* if this is the problem and I must start the original town all over - that's OK, but I want to make sure before I do that...