Hi,
The Fact the iPad can do it suggests the Apple ID is active on the iPhone.
It hints that the Messages app and the iMessages account on the Mac are not logged in (despite what the app might tell you).
Can you set the iPhone to Send from it's own Number.
Can you then send an iMessages to the Apple ID ?
Does it arrive on the Mac ?
If you deselect the iPhone Number on the Mac can you send from the Apple ID to the iPhone Number ?
I would try using the Sign Out button on the Mac and then Quirt the app before adding back the password and re-enabling the Account on starting it up again.
8:37 pm Monday; March 16, 2015
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