Tensor Operators

OP's candidate definition is a direct transcription of the tensor operator notion used in physics (and e.g. in Sakurai section 3.10) into a manifestly coordinate-independent mathematical construction. Tensor operators are e.g. used in the Wigner-Eckart theorem.

In this answer we suggest the following slight generalization of OP's candidate definition. Let the following five items be given:

1. Let $G$ be a group.

2. Let $H$ be a complex Hilbert space.

3. Let $\rho: G \to GL(V,\mathbb{F})$ be a group representation.

4. Let $R:G \to B(H)$ be a group representation.

5. Let $T:V\to L(H;H)$ be a linear map.

Definition. Let us call $T$ for a $G$-equivariant map if $$\begin{align} \forall g\in G, v\in V: &\cr T(\rho(g)v)~=~& {\rm Ad}(R(g))T(v)\cr~:=~&R(g)\circ T(v)\circ R(g)^{-1}. \end{align}\tag{*} $$

OP's candidate definition may be viewed as a special case of definition (*). For instance, if $\rho_0: G \to GL(V_0,\mathbb{F})$ is a group representation, then one may let $\rho: G \to GL(V,\mathbb{F})$ in point 3 be the tensor product representation $\rho=\rho_0^{\otimes m}$ with vector space

$$V~=~V_0^{\otimes m}~=~\underbrace{V_0\otimes \ldots \otimes V_0}_{m \text{ factors}}.$$

The definition suggested by joshphysics and clarified by Qmechanic already exists in the literature under then name of representation operator. This is discussed in, e.g., Sternberg's Group Theory and Physics, as well as the somewhat more elementary text An Introduction to Tensors and Group Theory for Physicists by Jeevanjee.