HI,
I personally would have deleted it as well.
I take it it was recreated ? (but that the iMessages account does not change behaviour ).
At present I see little the end user can do (further to what has been tried) on this issue.
If I was experiencing this myself and had tried the above I would then delete the Apple ID from all devices.
On the Mac I would delete the com.apple.imessages.bag.plist and the com.apple.ids.service.com.apple.madrid.plist one (delete the server references and the ID listings.)
On the iPhone and iPad I would remove the Apple ID and the iPhone Number (where required) and then turn off the App
I would then set the iOS devices to Airplane mode to ensure disconnect from the servers.
I would Quit the App on the Mac (if not done already.)
Following a restart of the Mac I would then set up the Apple ID there.
Then on just the iPhone I would enable Messages just for the iPhone Number.
I would then send a test iMessage from the Mac (Apple ID) to the iPhone (Number) and a return one from the iPhone Number to the Apple ID.
Only if this worked would I Add the Apple ID to the iPhone to create the Sync options and set up any other iOS device.
If it did not work I would consider visiting the Apple ID page and trying to re-verify the various items.
The fact that it is only the Macs that are affected makes it look like an Account/Keychain key issue.
Some of those things may not be changeable by the end user.

8:10 pm Thursday; September 18, 2014
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
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