HI,
Just to be clear.
Messages is the App name. iMessages is the new part (Compared to iChat) that has been added to communicate with iOS devices using Apple IDs and iPhone Numbers and is an "Account" within the Messages App.
You are showing us a Picture of a Buddy list (The iChat Style Account type in Messages).
This means you are not using a Google Mail ID that happens to be your Apple ID as the iMessages account name. You have been clear you are using as a Jabber Client.
Can you login to Google Mail via a Web Browser and text chat from there ?
Do you use this ID in the Mail App and does it work there ?
8:43 PM Tuesday; January 22, 2013
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously