What is the agreed upon definition of a "positive definite matrix"?

The positive definiteness (as you already pointed out) is a property of quadratic forms. However, there is a "natural" one-to-one correspondence between symmetric matrices and quadratic forms, so I really cannot see any reason why not to "decorate" symmetric matrices with positive definiteness (and other similar adjectives) just because it is in "reality" the form they define which actually has this property. I can see this one-to-one correspondence as one of the reasons why the symmetry should be implicitly assumed when talking about positive definite matrices.

One can of course devise a different name for this property, but why? In addition, positive definite matrix is a pretty standard term so if you continue reading on matrices I'm sure you will find it more and more often.

Some authors (not only on Math.SE) allow positive definite matrices to be nonsymmetric by saying that $M$ is such that $x^TMx>0$ for all nonzero $x$. In my opinion this adds more confusion than good (not only on Math.SE). Also note that (with a properly "fixed" inner product) such a definition would not even make sense in the complex case if the matrix was allowed to be non-Hermitian ($x^*Mx$ is real for all $x$ if and only if...).

Anyway, for real matrices, it of course makes sense to study nonsymmetric matrices giving a positive definite quadratic form through $x^TMx$ (which effectively means that the symmetric part is positive definite). However, I find denoting them as positive definite quite unlucky.

If the square matrix A is positive definite then the hypersurface defined by

$$Y=X^T.A.X$$

is convex.

Symmetric real-valued matrices are diagonizable, i.e., even when viewed as complex matrices their eigenvalues are all real.

For general complex square matrices replace 'symmetric' with 'hermitean'.