OP's candidate definition is a direct transcription of the tensor operatortensor 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 theoremWigner-Eckart theorem.
In this answer we suggest the following slight generalization of OP's candidate definition. Let the following five items be given:
Let $G$ be a group.
Let $H$ be a complex Hilbert space.
Let $\rho: G \to GL(V,\mathbb{F})$ be a group representationgroup representation.
Let $R:G \to B(H)$ be a group representation.
Let $T:V\to L(H;H)$ be a linear map.
Definition. Let us call $T$ for a $G$-equivariant mapequivariant map if $$ \forall g\in G, v\in V :\quad T(\rho(g)v)~=~ {\rm Ad}(R(g))T(v)~:=~R(g)\circ T(v)\circ R(g)^{-1}. \tag{*} $$$$\begin{align} \forall g\in G, v\in V: &\cr T(\rho(g)v)~=~& {\rm Ad}(R(g))T(v)\cr~:=~&R(g)\circ T(v)\circ R(g)^{-1}. \end{align}\tag{*} $$
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
$$V~=~V_0^{\otimes m}~=~\underbrace{V_0\otimes \ldots \otimes V_0}_{m \text{ factors}}.$$