lagrangian formalism - Peskin and Schroeder passive and active translation

In peskin and Schroeder's qft book, in chapter two, they're discussing Noether's theorem with respect to translations of co-ordinates.

They describe and "infinitesimal" translation $x^\mu\rightarrow x^\mu -a^\mu$.

And say that as an alternative it can be seen as a transformation of the field configuration as

$\phi(x)\rightarrow \phi(x+a)=\phi(x)+a^\mu \partial_\mu \phi(x)$.

Now according to David tong's notes this is the active point of view of the transformation but I'm still a bit confused by either viewpoint. I've read posts here on physics stack, which are about the same thing but they haven't helped me so far.

The way I see it if we set $f(x)=x-a$ Then $\phi'(x)=\phi(f^{-1}(x-a))=\phi(x+a)$. Is this the idea of an active transformation?

In which case would the passive version simply be $\phi(x-a)$?

I've read the example in boas with SO(2) acting on $\Bbb{R}^2$, where they said in one viewpoint we change the basis but in the other viewpoint we change the vector itself. Is there any benefit to choosing either though in the above case of peskin and Schroeder, wrt it's context of Noether's theorem?