From Wikipedia:
Sylvester's criterion states that a Hermitian matrix ''M'' is positive-definite if and only if all the following matrices have a positive determinant:
- the upper left 1-by-1 corner of $M$,
- the upper left 2-by-2 corner of $M$,
- the upper left 3-by-3 corner of $M$,
- ...
- $M$ itself.
In other words, all of the leading principal minors must be positive.
So do we call a principal minor positive if it has a positive determinant? Because from my knowledge a positive matrix is just a matrix with positive elements. In this sense the positive definite hermitian (symmetric) matrix
$$\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$$
would have principal minors which are not positive matrices.