How do I compute the following gradient? $$ \nabla_{\phi} \,\,\,T(y, x(\phi, y), \phi) $$ Should I use the product rule or the chain rule? Is it as follows?
$$ \frac{\partial}{\partial x}T(y, x, \phi) \frac{\partial x(\phi, y)}{\partial \phi} + \frac{\partial}{\partial \phi} T(y, x(\phi, y), \phi) $$
My primary interepretation would be that $\nabla_{\phi}$ refers to the partial derivative with respect to the 3rd argument, hence $$\nabla_{\phi} \; T(y,x,\phi) = \frac{\partial}{\partial \phi} T(y,x,\phi),$$ which is the second term in your answer. That is, $\nabla_{\phi}$ refers to the partial derivative of $T(y,x,\phi)$ with respect to argument $\phi$, not the total derivative of the variable $\phi$.
However, if you let $H(\phi, y) := T(y, x(\phi, y), \phi)$, then
$$\nabla_{\phi} \,\,\,H(\phi, y) = \frac{d}{d\phi} T(y,x,\phi) = \frac{d}{d\phi} T(y,x(\phi, y),\phi) = \frac{\partial T(y,x,\phi)}{\partial \phi} + \frac{\partial T(y,x,\phi)}{\partial x} \frac{\partial x}{\partial \phi},$$ which is what you wrote. That is, the total derivative with respect to $\phi$.
So I would say it is unclear what the notation refers to, although I would think it refers to the former method. To my experience, when the notation is $\nabla_{\boldsymbol v} \; T(y,x,\phi)$, with bold subscript, $\boldsymbol v$ refers to a vector and the expression refers to the directional derivative.