Theorem 1 Let $X$ be a normal integral scheme, $K$ its function field, $\bar{K}$ an algebraic closure of $K$, and $M$ the composite of all finite separable field extensions $L$ of $K$ with $L\subset K$ for which $X$ is unramified in $L$. Then the fundamental group $\pi(X)$ is isomorphic to the Galois group $\operatorname{Gal}(M/K)$.
In the above paragraphs Theorem 1 at the beginning of this post is being used to compute the etale fundamental groups of $\mathbb P_{K}^1$ and $\mathbb A_{K}^1$.
I've difficulty in understanding the difference between the two cases.
In particular, I'm particular I've difficulty in understanding how to figure out if a finite extension of $F$ of $k(t)$ is unramified over $\mathbb P_{K}^1$ or $\mathbb A_{K}^1$?
And in case $\mathbb A_{K}^1$ are we looking only at valuations $v_f$ (and not $v_{\infty}$) and checking if $F$ is unramified at all these valuations?
In case of $\mathbb P_{K}^1$ are we looking at all the valuations $v_f$ and $v_{\infty}$ and checking if $F$ is unramified at all these valuations? Is that the difference? Why is checking at $v_{\infty}$ and $v_f$ enough in this case?