Let $X$ a functional space space (I mean a space whose elements are functions). Let $f \in X$. Say I define these two operator (argument scaling and argument translation as follows)
$S_\alpha$ is a scaling operator means that $$\forall f \in X, S_\alpha f = g, g(x) = f(\alpha x).$$
$T_{x_0}$ is a translation operator means that
$$ \forall f \in X, T_{x_0} f = g, g(x) = f(x - x_0).$$
I think that notation is correct so far. Would be correct to write
$$ \begin{array}{l} S_{\alpha} f = f(\alpha x) \\ T_{x_0} f = f(x - x_0) \end{array} $$
Or shall I write something like
$$ \begin{array}{l} \left(S_{\alpha} f \right)(x) = f(\alpha x) \\ \left(T_{x_0} f \right)(x) = f(x - x_0) \end{array} $$
Also for the derivative
$$ \frac{d}{dx} f = f^{(1)}(x) $$
or
$$ \left(\frac{d}{dx} f\right)(x) = f^{(1)}(x) $$
(Maybe the last one can sound silly, but sometimes I'm puzzled in the meanings)
$S_\alpha f$ is a function. It is defined by the expression for its values at an arbitrary $x$, namely $(S_\alpha f)(x) = f(\alpha x)$. The function $S_\alpha f$ should not be confused with $(S_\alpha f)(x)$, which is merely its value at $x$. You argue similarly for the correct notation in all the other definitions.
It's no different from the distinction we make with any other function and its values. For example, the exponential function $\exp:\mathbb{R}\to\mathbb{R}$ is not the same as $\exp(x), x\in\mathbb{R}$: the first is a function $\mathbb{R}\to\mathbb{R}$, the second is just an element of $\mathbb{R}$ (provided that $x$ is given). The tendency to identify the function $\exp$ with its values $\exp(x)$ is merely an abuse of notation.