Start here: this is the operation marketed as a BBL
Buttock fat transfer and a Brazilian butt lift are the same operation. Gluteal fat grafting is the clinical name; BBL is the marketing one, and nothing about the technique, the risk, or the genuine price changes according to which phrase a clinic prefers to use.
Using the clinical term tends to produce a more precise consultation, because it directs the conversation toward harvest technique, injection plane and realistic volume rather than toward a branded package built around a headline figure. The questions that determine safety and outcome are identical either way, and this page addresses them using the clinical framing.