I don’t know much about how live query works on the server side to answer what you should do there. My guess is that once a connection is closed the server will discard the subscriptions, but @davimacedo might have more info here.
I’ll point out that now on the client side, after you unsubscribe from all of your subscriptions, it will automatically close the connection.
The second and third way you mentioned seems reasonable to me. It just matters your scenario. If you are subscribing to a new query in scenario 2, it should reconnect and also resubscribe to any previous queries as well. Scenario 3 should reconnect and resubscribe to all previous queries.
Note that you can also use: ParseLiveQuery.client?.open, ParseLiveQuery.client?.openPublisher, ParseLiveQuery.client?.close(), ParseLiveQuery.client?.closeAll()