I'm guessing the Cloud DVR box and the Wireless 4K Contour Stream Player have a different Program Guide than the traditional cable-boxes (coaxial). I have a traditional box and have never seen programs marked as "Internet" in its Program Guide.
However, programs marked as "Internet" are for watching a program via the Internet (streaming).
Meaning: Source > Internet > Cable Box.
However, we're talking about recording a program marked as "Internet" to watch later.
Meaning: Source > Internet > Cloud DVR Server > Cable Box.
I should be charged once from Source to the Cloud DVR Server, but what about being charged...umpteen times...from Cloud DVR Server > my Cloud DVR Box?
www.cox.com/.../data-usage-and-cox-services.html