More specifically, I see $\frac{\partial}{\partial x}f(x,g(x))$ as meaning take the derivative of $f(x,g(x))$ w.r.t $x$.
On the other hand, I see $f_x(x,g(x))$ as meaning the partial derivative of $f(x,y)$ evaluated at $(x,g(x))$
Do these notations have clearly defined meanings -- i.e. is it clear whether we take the derivative and then evaluate or evaluate and then take the derivative -- or does it depend on context?
If they do not, how can I be clear in which order to take the derivative and evaluate?
As an illustration, consider $f(x,y) = x^2+y^2$
Then $f_x(x,y) = 2x$
I would then say that $f_x(x,g(x))= 2x$
Whereas I would say that $\frac{\partial}{\partial x}f(x,g(x)) = \frac{\partial}{\partial x}\left(x^2 + g(x)^2 \right)= 2x + 2g(x)g'(x) = f_x(x,g(x) )+f_y(g(x))g'(x)$
You have to distinguish between $\frac d {dx}$ and $\frac {\partial} {\partial x}$. Your calculation of $\frac {\partial} {\partial x} f (x,g(x))$ is wrong and you have to intepret it as $f_x$ evaluated at $(x,g(x))$.