Hi,
To be honest it sounds like you.
You will notice that in System Preferences > iCloud that there is no setting to Enable Messages (or FaceTime the App Store or Tune or iBooks).
However when you open Messages and then Preferences > Accounts the Apple ID you use in the iMessages account must be an Apple ID one.
Most would set it up this way:-
Enable the Messages app on the iPhone in Settings > Messages and wait from the number to Verify.
On the Mac version enter the Apple ID you are going to use.
Once that is set to go it will list just the Apple ID in the Receive at list.
On the iPhone enter the Apple ID or use the Contact app to select yourself and that part of your ID set up.
Send An iMessage to the Apple ID (The Mac only at this stage)
Return the iMessage (Apple ID to iPhone Number).
If that works add the Apple ID (the same one - not ones that might be linked to the same iCloud Account) to the setting for Messages on the iPhone.
Set which ID is going to be uses for "Start Conversations From".
This should prompt a pop up on the Mac saying what the iPhone is using.
Accepting this enables the iPhone Number on the Mac Receive At list and shows you the Start Conversation From Option drop down.
At this point iMessages to either the iPhone Number or the Apple ID should appear on both devices.
If you use different Apple IDs (and iMessages considers linked IDs in iCloud to be different IDs) then the process will not work.
You can end up with Linked ID if for instance you had an @mac.com ID you used in MobileMe as a paid for account which also gave you an @me.com ID and then Linked them to iCloud.
This is effectively three Apple IDs on one iCloud Account.
As you add the Apple ID to more devices your should accept the Pop ups even if the Apple ID is the same.
The picture tell us nothing accept you have either had no Messages from any service or you have cleared the list on the left.
As the "To" Spot has not entry the app has not decided which Account you are using (although the lack of Accounts listed in the lower left corner does suggest only the iMessages might be active)
In the case of Messages the "Sync" means Display on all Devices. It also only applies from the moment you set it up.
It will not play catch up.
iMessages are only on the iMessages Servers until they are delivered.
Effectively it "Copies" each iMessages and Pushes it at each device uniquely and deletes once that device accepts it.
Once all devices have "joined" then they will show the same iMessages but only from that point on.
IF I clear my list in Messages and then select New Message like you have and have the cursor in the "To" spot there is no identifying "name of service" in he text field.

However all the logged in (Buddy List ) accounts are scrollable in the left hand corner. (iMessages does not show here).
The Push that iMessages uses is not the same as the way the Mail App "Fetches" new emails.
Emails can be left on the server (or stay on the server in the case or IMAP servers) and are "fetch-able" by any fresh login to those servers with your IDs.
To be clear:
The Apple ID site https://appleid.apple.com/uk/ is separate from the iCloud service https://www.icloud.com/
iMessages is not part of iCloud (neither is iTunes, the App Store, FaceTime or iBooks)
The devices have to use the Same IDs to "sync"
They only then do so from that start point onwards.
The "Sync" is only Display on All Devices" and anything you do to one iMessage on one device does not effect other devices.
The iMessages are deleted from the servers once delivered.
They are also encrypted in transit (not even Apple can read them) .
From what you post I am not even sure you are using the same ID on each device.

9:34 pm Sunday; January 10, 2016
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