A self-adjoint linear operator $\tau$ is referred to as positive if is associated quadratic form $\langle\tau v,v\rangle\geq 0$ for all $v$, and is referred to as positive definite if $\langle\tau v,v\rangle>0$ for all $v\neq0$.
There are also theorems stating that $\tau$ is positive if its eigenvalues are nonnegative, and is positive definite if its eigenvalues are all positive.
Any reason why this is instead of calling $\tau$ nonnegative/positive? That nomenclature seems like it would make more sense to me.
The terminology makes the sense that you expect as an order on the operators. $A$ is positive if it is not the zero operator and it is positive in the sense you defined. $A\geq B$ iff $A-B$ is positive. Every linear operator vanishes at zero, so the definition has to reflect that.
Definite is the word that is strange. But it just means that zero is the only place where the operator vanishes.
A particular case: Consider linear operators in $\mathbb{R}$ as $\mathbb{R}$-space. These are of the form $f(x)=ax$ for some $a\in\mathbb{R}$. When is $f$ positive?
In this example it is definite the concept that is not very relevant. But when the dimension increases then it becomes important.