Yes indeed. I said "private" to give the reader an image somehow that is separated from the well known "public".
Literally a mesh network with local nodes.
I've seen many noobs saying that in order to use BTC over LN they have to be connected to a well known node. That is false. You can still transact over LN with just 2 nodes. But yes, only between them.

Lets say the nodes in a village will be hyper connected between themselves or have one big bank in between, and that bank will have connectivity to the world

Exactly that scenario I described in this guide.