Although I’m on your side here, the opposition is formidable.
Limitation by association with quadratic forms goes back at least to Gantmacher (1950’s). Springer’s Encyclopedia of Mathematics is in this camp. Gil Strang has promulgated the “restricted” version in his textbooks, which may have poisoned influenced the Julia developers.
Most others (e.g. Golub & van Loan, LAPACK, Mathematica) are careful to write “symmetric (or Hermitian) positive definite” where that applies.
Since it seems obvious to many of us that definiteness is a feature of bilinear forms without regard to symmetry, the LinAlg documentation should be clarified if your suggestions are not followed. Or maybe there should just be separate functions: ishpd, isspd, isposdef?