HI,
Quick revision.
On the iPhone go the Settings Messages and Enable the app.
Also Enable the SMS option.
Make sure in the WiFi setting the iPhone is on the same LAN as the Mac.
ON the Mac Add the Apple ID to the iMessages Settings in Messages > Preferences > Accounts.
Start a New Messages on the Mac and manually type the iPhone number in full including Country Code.
Does the iPhone get the iMessage ?
If so add the Apple ID to the Send and Receive part of the Setting on the iPhone.
At this point both devices can send and receive on either ID or iPhone Number.
If the Wifi for the iPhone has been done then the Text Forwarding can be turned ON
This create the Code Number on the Mac that has to be entered back on the iPhone.
If the Mac is getting and sending SMS then the issue is something that has happened after the set up above.
This can include the Connection to the iMessages Servers by the Mac failing for some reason and then you will only see SMS but not iMessages.
This will deteriorate as the link between Apple ID and iPhone degrades due to the Apple ID not logging in which leads to the sync failing.
Do this (When the iMessages account will not Login) if the Mac is not sending iMessages as it is not connected to the Server and the chances are your Appel ID when use on that Mac for iMessages is blocked (Other logins and services will not be effected).
8:14 pm Monday; May 28, 2018
iMac 2.5Ghz i5 2011 (Sierra)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone 6 iOS 11.x and an iPad (2)