Is it therefore, incorrect to talk about "expectation value of an operator"?
Yes, because when you write those integrals you're asking for the average of a dynamical variable $A$ whose associated operator is $\hat A$. The point is that you're asking for a number , e.g. the average position of the electron in a gaussian wavepaket. Then, when you want to calculate this expectation value $\langle x \rangle$, the associated operator shows up in the integrals. And it's not only semantics: what's the use of asking, let's say, "hey, what's the average of the Hamilton operator?" ?
This would mean that you had a bunch of Hamilton operators and you wanted to know the average of those operators ... This makes no sense. ^^
You could ask for the average energy of a system: then you would use the Hamilton operator in the integrals you wrote.
So the answer to your second question (Is it incorrect to write ...) is simply yes.
(This stuff is confusing...)
I just opened my Shankar and came across this:
See equation 7.3.2.: he is looking for the mean energy $\langle E \rangle$. So according to my reasoning, he writes $\langle \Psi|H|\Psi \rangle$. But then he writes $...=\langle H \rangle$, which confused me a lot. After some thinking I came up with the solution: here, he treats $H$ as an observable. And the way he treats the Hamiltonian the next section proves this: the momentum and position are treated like normal observables and not like operators.
Then, the next page, he writes this:
So here he uses $\langle H \rangle$ as a synonym for the mean energy.