These principles became part of pilates in 1980, long after Joe died, in the book by Friedman & Eisen, Chapter 3 The Pilates Method of Physical and Mental Conditioning. If you look for these pivitol principles, listed as such before this book, directly from Joe, you're gonna be looking for a long time.
When you refer to these principles it seems obvious you would refer to control first because it is the name of his method "Contrology". Then concentration because of Joe's assertion of it primary importance, the others, center, precise flowing movement riding within the breath are fine with me.
"Pilates is so much more than concepts, but keeping these 6 principles in mind has been key to real progress for many Pilates students." Sure this is true, but it avoids what pilates could be that is more than concepts, and relies on a second generation synthesis into principles to offer meaning to something that is clearly much more than that.
Swim upstream, look for original material, make your understanding from there.