What is the proper procedure to update the client on Mac? Is it necessary to download a fresh client from the UrBackup server? Is there an easier way, and when 2.x is out of beta, will the Mac client automatically update like the Windows clients do?
It should already autoupdate. I have only tested the silent autoupdate, however. And you need relatively recent versions, e.g. 2.0.9=>2.0.10.
My two Macs are on 2.0.5. So if I reinstall the backup software, they will auto-update from here on?
Yes, should. If you want to test you can install 2.0.9 
Strange. Today I updated my test server to 2.0.11 and I downloaded the Mac installer from the web interface to update this machine and it installed 2.0.6. Is this expected behavior?