I take Toshiro Daigo's book Kodokan Judo Throwing Techniques to be the official word on throw classification. Daigo is the chief instructor at the Kodokan, the mother school of judo in Tokyo, Japan.
obi-tori gaeshi (sutemi version) is hikkikomi gaeshi
The top player in the lightweight class was a man named Stepanov, who inflicted pain on many Japanese judoka, with ashi-tori-waza from Sambo Judo, and unorthodox techniques such as obi-tori-gaeshi.
The technique, in which he gripped the back of the opponent's belt and dropped down the rear, has long been called obi-tori-gaeshi, but is in fact the hikkikomi-gaeshi described in sono ni.
"Sono ni" is the second variation/point of the section on hikkikomi-gaeshi, where an over the back grip is taken on the belt.
I think the quotation means that all obi-tori-gaeshi variations are hikkikomi-gaeshi, but it's not totally clear. Hikkikomi-gaeshi is one of the 67 Kodokan-recognized throws, and obi-tori-gaeshi is not.
Obi-tori-gaeshi is on the list of now 68 Kodokan-recognized throws as a te-waza. See john's answer for description and video.
Differences between hikkikomi-gaeshi and sumi-gaeshi
Daigo explicitly enumerates points that differentiate variations of hikkikomi-gaeshi and sumi-gaeshi, so you are in good company regarding the confusion between the techniques.
Without over the back grip:
The key point in telling these techniques apart lies not in the part of uke's leg that tori springs upwards, but in whether he executes the technique from a defensive grappling posture (sumi-gaeshi), or executes it while holding one of uke's arms (hikkikomi gaeshi).
With over the back grip:
The key point for telling these techniques apart lies in whether tori grips uke's back (sumi-gaeshi) or whether he grips the back of uke's belt (hikkikomi gaeshi).
Judo throw classification is a messy business. Throws may have different names at different times, and the points of classification can be quite arcane and perhaps not useful. The difference between sumi-gaeshi and obi-tori-gaeshi is not something I actually remember, but I know where to look it up. I would not expect most instructors to know or care about this difference.