Umm,
I think you have all the info in the previous Posts but may be a run though might help.
I have not spotted if you have said the other person is using an iPhone.
Lets start with your iPhone.
You can turn the app On and it will, after verifying the iPhone Number (Register) be able to send iMessages to iPhone Numbers (that are also registered) or to Apple IDs that are registered.
Apple IDs can be used on Macs, iPads and some iPod Touch devices.
Then you can elect to turn On SMS when the iPhone will send SMS to non iPhones or to iPhone that are not registered with iMessages (as a service) when the app thinks it cannot send an iMessage.
So already you have lost some control as the app decides if the iPhone at the other end can receive iMessages.
Whilst the other end might be registered it might not be able to Login due to the wrong type of Carrier Service, lack of WiFi service or simply being without power.
All of these may generate an Error messages that reads "Not Registered with iMessages".
The Mac.
Next you can add/register an Apple ID on your Mac.
This allows that device to send iMessages to any ID that is registered be it Apple ID or iPhone Number.
The Mac version cannot be set to send SMS from the Mac version - it requires the steps that follow.
Linking Apple ID and iPhone Number for iMessages
Next in Send and Receive on the iPhone > Settings > Messages > Send and Receive you can add the Apple ID.
This causes a Pop up on the Mac version that you should accept.
This Links the Apple ID on the Mac and the iPhone Number on the iPhone.
On the iPhone both are seen in Send and Receive.
On the Mac version they both Appear in the Receive At list and make the "Start Conversations from..." drop down to appear.
On both devices you can then set the ID for Start Conversations From.
It can be the same on both of they can remain the lead ID fro each device (iPhone Number on iPhone and Apple ID on Mac).
There can be issues though if you deselect either ID from the Receive At or Send and Receive options.
his can break the link between iPhone Number and Apple ID.
Linking the two devices for SMS.
This requires that you have done all of the above steps.
You also need to have the iPhone join you Network, normally by WiFi.
Then you go to Text Forwarding on the iPhone and turn it On.
This should generate a 6 digit code that appear on the Mac.
Entering that code back on the iPhone will link the two for SMS.
You can add other devices by going back to Text Forwarding and picking the other devices and entering the code that they produce.
This makes the iPhone "Share" it's connection to the Carrier over your LAN/WiFi network.
The Mac should see SMS messages arriving on the iPhone and be able to Send replies from the Mac.
You may find that the first reply may need to be Sent for the iPhone.
Here again the iPhone decides if it can send the SMS even if it is passing on what came from the Mac.
I do realise that this might come across as trying to teach a grandmother to suck eggs but sometimes laying it out as simply as possible seems to be the way to go.
10:02 pm Thursday; March 1, 2018
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 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)