You should be very very careful with field transformations. In a quantum field theory, only the field transforms -- coordinates DO NOT TRANSFORM, even for spacetime transformations. However, describing spacetime transformations as acting on the coordinates is often a useful tool to describe how the fields themselves transform. But you must always remember that at the end of the day, the action on the coordinates is a tool.
Every single symmetry transformation is described by a field transformation
$$
\phi(x) \to \phi'(x)
$$
To describe the symmetry, we must then explain what is $\phi'(x)$ in terms of $\phi(x)$. For instance, for a $U(1)$ gauge transformation, we write
$$
\phi'(x) = e^{i Q \alpha(x)} \phi(x)~.
$$
When the field transformation corresponds to a spacetime transformation, it is convenient to describe $\phi'(x')$ as opposed to $\phi'(x)$. For instance, for scalar operators, we write
$$
\phi'(x') = \phi(x)~.
$$
Note that this is just a tool for us to determine what is $\phi'(x)$. For instance, for translations, we have $x'=x-a$ so that
$$
\phi'(x') = \phi'(x-a) = \phi(x) \quad \implies \quad \phi'(x) = \phi(x+a)~.
$$
For Lorentz transformations, $x' = \Lambda x$ so that
$$
\phi'(x') = \phi'(\Lambda x) = \phi(x) \quad \implies \quad \phi'(x) = \phi(\Lambda^{-1} x)~.
$$
