My guess that apple support says it is fine, cause your authentication for both services is the same. What suppose to happen though is emails go to your mail app. Imessages go to your messages app. My guess would be that you are using same id for the message on your macbook and ipad. Now you are getting mixed up, cause only one device sometimes gets the message intended for you. Sometimes both, but there is no consistency. If my guess is correct it is unfortunate, but really is how imessage behaves. It does not have a mechanism to be distributed to all devices, like e-mail does.
I think I understand what you are saying. What I dont understand is why this wasnt thought thru more when for Mountain Lion, it was decided bu Apple, to add the iMessage feature. And then, the fact that they hoped many of their customers would own both the iPad and a Mac.
But I would gladly turn off the feature on my MBP for iMessage, if I could get it to work right again on the iPad. I am going to make another call to apple support to let them know that it still doesnt work (their only offer on our call yesterday was to log off/back in to my apple id. Then I had to test it later when one of my friends was available to iMsg. But since my ipad is past the phone support time, it could be difficult, IDK, to get them to help me turn off the MBP imessage function in hopes that that would then work right again in the iPad.
Using FaceTime http://support.apple.com/kb/ht4319
Troubleshooting FaceTime http://support.apple.com/kb/TS3367
The Complete Guide to FaceTime + iMessage: Setup, Use, and Troubleshooting
Troubleshooting FaceTime and iMessage activation
Using FaceTime and iMessage behind a firewall
iOS: About Messages
Set up iMessage
Setting Up Multiple iOS Devices for iMessage and Facetime
FaceTime and iMessage not accepting Apple ID password
Unable to use FaceTime and iMessage with my apple ID
For non-Apple devices, check out the TextFree app https://itunes.apple.com/us/app/text-free-textfree-sms-real/id399355755?mt=8