Considering finite dimensional spaces: Positive operators are a subset of hermitian operators. They have a matrix representation. Therefore one often uses both terms (the operator and the matrix representation) interchangably (e.g. in Quantum Mechanics). Is said matrix called "positive" (just like the operator) or "positive semidefinite" (which would be usual when only considering matrices)?
The definition of positive operators I use: A hermitian operator $A$ is called positive, if for any vector $|\psi\rangle$, the scalar product $(|\psi\rangle,A|\psi\rangle)$ is a real,non-negative number.