Every TV will work exactly the way you're thinking. I have had the X1 DVR for years, and what I explained is what it does. The correct unit is the one with the blue power light on the front right corner of the unit. I have one right here. XiD-C is the model number - unless there is a newer one. See below.
I've had mixed results with the call center. I usually go to the Xfinity store and talk face-to-face with a person. You can walk out with the correct equipment.
Most recently I have received excellent help from the Xfinity social media team. Reach out to "ComcastElla" here. She's awesome.