It's not so much that one version is the correct one and the others are incorrect or watered down versions, it's more a case of word usage:
If someone pauses and then continues, they may well quite rightly call it Rest-Pause which to all intents and purposes it is; it may well be that one version was logged, by a trainee, and others began to follow the same style of set-up calling it Rest-Pause as they went along which would eventually see it named the same way as the inital traineee did...that's how set-ups and training styles originate.
There are so many ways to Rest-Pause a set but Rest-Pause is just an all-encompassing term, none of the set-ups, no matter how many there are, are either right or wrong, they're simply named appropriately.
Eventually, it may be that people will use other additional words to distinguish one style from the other such as with DLs, of which there are many variations, none of which are wrong; sometimes, it's just a matter of terminology.
36.5 kg /80.3 lb Middle-Finger DL (right hand)...