Hi,
We do know from trial and error and comparing the issue with other Login issues that there are situations where for some reason the Apple ID when used on the Mac does not login or not login properly.
It starts as it were with this Error message.

People will report this as only effecting the Mac and that the Apple ID "works everywhere else".
It can only be solved by Contacting Support as it says.
When people do this - sometimes bumping it up to a level 2 person - that the Apple tech says they made changes to the Apple ID to get the iMessages server to respond to it.
It then seems there is one where the app (on the Mac) looks like it has logged in but all the Contacts get the "Not Registered with iMessages" Alerts.
It seems this is close the the above issue and requires an Apple tech to intervene.
I have seen this one where iMessages get the red " Not registered" alert but SMS text can still be sent (which Uses the iPhone's Carrier) which puts it back on the state of the Apple ID login.
Of course there are option with "Start Conversations From" but I have not been able to tie down if this makes a difference to how the Mac responds or it's ability to send iMessages.
I expect that it will come down to the Apple ID login rather than being able to use the Start From options to get around it.
The Text Forwarding is similar to Paring in Blue Tooth but does not make use of Blue Tooth at all.
It requires both the Mac and iPhone are on the same network (it says WiFi in the blurb but the Mac can be Ethernet) and that the iPhone has Carrier service.
I would suggest contacting Apple Support.
You may find you have to go to Level 2 as Level 1 people are somewhat Script led and expecting Software and Hardware issues (fixing Account/Apple ID issues are free).

10:09 pm Thursday; December 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