Where are playlists stored

We have a setup with Vault (original version) and a number of players (from original to 2i).

As part of a support request about one player working we had a back-and-forth with their support and it seems that they were unable to tell me where our playlists (NOT the SONGS but the PLAYLISTS) are actually stored. I always thought they were stored centrally on the Vault.

Support said that they are stored locally on the player--but that does not make complete sense, because any playlist I create is available on ANY of my networked players. On the other hand it does seem like, when I move one player to a new network, SOME of the playlists are still on the player (even if much of the music it references is not there, because it lives on the Vault)

So: Do playlists automatically get replicated across players?
If so, what happens if I edit a playlist on one player while in the network--do the edits get replicated across all players, or do I now have playlists with identical names that are different across players?
What if I edit the playlist on a player while it is NOT on the network?

This is very confusing. It is extremely important to us that when a playlist is changed, it is changed for ALL players, not just the one I may have currently selected in the controller.

Does anyone know how this architecture really works?