moving files from one network folder to another--does the data come through my local computer?

Solution 1:

The 1st one is just a rename in the same share so that is nearly instant and done server-side.
Both others go through your local computer. (Data will traverse the LAN twice.)

This is actually not always true: If the servers involved are Windows 2008 or Windows 2012 and the server administrator has configured some special handling and your local PC is at least Windows 7 then it may be possible that the move will be done server<->server.
(This feature is available as of Server 2008, but I have not seen it really being used anywhere.)

In order to have to move done as efficiently as possible you will have to logon on the server itself (either on the console or via RDP) and do the move from there.