HBO is iconic. You wouldn't rename Apple - Mac. Starbucks, Star.
Don't change an iconic name to something lesser by picking Max. Nobody cares about Cinemax. Nobody. Everyone knows HBO. It's not TV it's HBO. Home Box Office
When you saw these as a child/adult, you knew what was coming next. Quality, an amazing show/movie with no commercials!
Family time.
You wouldn't rename "Federal Express" to "FedEx" or "International Business Machines" to "IBM" or "United Parcel Services" to "UPS" or...