The Stinespring dilation shows that there is a compression like
$\varphi(a)=P_{H}\pi(a)|_{H}$
in which case we have a restriction of $\pi(a)$. In several other places I see the following notation
$\varphi(a)=P_{H}\pi(a)P_{H}$
in which case we use a projection on the right instead of a restriction.
Now I have a hard time to see that this is the same thing. It seems to be trivial since I can't find anyone writing about this. Same thing happens also when I look at the definition of the Toeplitz operator which is also sometimes defined using restriction and sometimes using a projection on the right side of the function. Having a projection on the left makes sense to me "visually" - if I apply a projection on a function I want to do with the output of the function but the other way around doesn't make sense to me. Can anyone explain to me why this is equvialent?
Per Martin’s suggestion, I’m transferring my comment to an answer:
Technically speaking, they aren’t the same since $P_H\pi(a)|_H$ is defined on $H$ while $P_H\pi(a)P_H$ is defined on the larger Hilbert space (on which $\pi(a)$ is defined). They are similar enough though - $P_H\pi(a)P_H$ is just $P_H\pi(a)|_H$ on $H$ and 0 on $H^\perp$. There are situations in which you technically should use one of them instead of the other, but since they are similar enough sometimes people just abuse notations and use them interchangeably.
Also per Martin’s suggestion, I should note that the Stinespring theorem applies to all cp maps, instead of just ucp maps. Personally I think there’s less to be confused about for general cp maps, since in that case the map $v: H \rightarrow K$ from $H$ to the larger Hilbert space $K$ (on which $\pi(a)$ is defined) is no longer necessarily an isometric embedding, so it doesn’t make sense to consider $H$ as a subspace of $K$, and as such, there’s no projection $P_H$ nor restriction of $\pi(a)$ to $H$. Instead, the Stinespring theorem gives $\varphi(a) = v^*\pi(a)v$. It just so happens that for ucp maps, $v$ is an isometric embedding, so one can consider $H$ as a subspace of $K$, $\pi(a)v$ as the restriction of $\pi(a)$ to $H$, and $v^*$ to be the orthogonal projection onto $H$.