Actually the term prep school tends to be in reference primarily to day schools (think Dalton, Harvard Westlake), or parochial prep (Regis, Delbarton) or public Prep (Bronx Science, Boston Latin), while the seven schools you reference are representative of boarding schools (with small numbers of day students).
I also question whether it appropriate to equate prestige with the term “best” as you seem to want to do. Elitism, selectivity or even reputation are trumped by fit when determining best.
Your terminology suggests a false hierarchy. I say this having attended one of the schools upon which you ascribe such lofty accolades.
The OP should not seek a school (if possible) solely based upon a perception of exclusivity. I recognize her parents are placing pressure on her and wish her the best in finding a school that checks all the requisite boxes as she looks to be an outstanding candidate.