That web page has some erroneous information on it (as do many others I've seen). No state should require a genuine CDL to drive any RV for personal noncommercial use, as what constitutes a commercial vehicle for CDL purposes, as well as the various classes of CDLs and similar base requirements, are all defined at the federal level. Recreational vehicles are specifically exempted as not being commercial vehicles when driven for personal, noncommercial use.
This is a good thing, as a CDL carries with it a considerable amount of administrative overhead and additional regulations: mandatory rest periods and logging of the same, periodic random drug tests, annual pysical exams, and so forth. That's in addition to the training and testing of actual driving skills.
Several states do require non-commercial licenses different than those for standard passenger cars, at least for large RVs, and often but not always the classes of these non-commercial licenses parallel those of CDLs (i.e. they take effect at the same vehicle weights). The best guide is to check with the DMV or equivalent for your state. If your license covers the vehicle in the state it's issued in, you can use it in any other state legally; though that doesn't guarantee that the vehicle setup itself would be legal in all other states. One example of this is double towing: in some states it's simply not permitted at all.
