Are Riemannian manifolds only referred to as diffeomorphic if the diffeomorphism is an isometry?

Question

Let $M$ and $N$ be Riemannian manifolds. My understanding is that strictly speaking, a diffeomorphism $\phi:M \to N$ only acts on the smooth manifold structure, not the metric tensor. But there is a natural action of $\phi$ on the metric tensor given by the pullback under the diffeomorphism, which in fact makes the diffeomorphism an isometry. If no diffeomorphism between $M$ and $N$ is an isometry, is it standard to still refer to the Riemannian manifolds as "diffeomorphic"?

Answer 1

The answer to your last question ("If no diffeomorphism between $M$ and $N$ is an isometry, is it standard to still refer to the Riemannian manifolds as "diffeomorphic"?) is Yes. (As @TedShifrin pointed out, this is at odds with the question in your title, to which the answer is No.)

To say that two smooth manifolds are diffeomorphic means there exists a diffeomorphism between them, irrespective of what that diffeomorphism does to any metric. If the manifolds are also endowed with Riemannian metrics, we say that they are isometric if there exists a diffeomorphism between them that pulls one metric back to the other.