If $y \in \mathbb P^1$ is a (closed) point and $V$ is an affine n.h. of $y$, then we may find a function $a \in \mathcal O(V)$ which vanishes precisely at $y$. If we let $U = f^{-1}(V)$, then $U$ is an open set containing the fibre over $y$, and the fibre over $y$ is cut out by $f^* a \in \mathcal O(V)$. Thus this fibre is a local complete intersection, and in particular Cohen--Macaulay, and in particular $S_1$.
Now let $\sigma$ be the section of $f$. Since $f\circ \sigma = \text{id}_{\mathbb P^1}$, we see that $f$ induces a surjection from $T_{\sigma(y)}X$ to $T_{y}\mathbb P^1$, i.e. (in differential topology language) $f$ is a submersion at $\sigma(y)$,
or in algebraic geometry language, $f$ is smooth in a n.h. of $\sigma(y)$. In particular, the fibre over $y$ is then smooth in a n.h. of $\sigma(y)$, and in particular, is reduced in a n.h. of $\sigma(y)$.
Thus this fibre, being irreducible (by assumption) is generically reduced.
A general theorem says that (for Noetherian rings, or equivalently, locally Noetherian schemes) being $R_0$ (i.e. reduced at all generic points) and $S_1$ is equivalent to being reduced. This applies here to let us conclude that the fibre over $y$ is reduced.
$\newcommand{\Spec}{\mathrm{Spec}}$
Scholze---> Katz-Mazur: I really wouldn't stress too much about this, to be honest. Probably Scholze should say that $p$ is locally of finite presentation and/or $S$ is locally Noetherian. Since the moduli spaces of such objects constructed is locally Noetherian, you really have no harm restricting to such a thing. Then, proper implies finite type and since S is locally Noetherian this implies that $p$ is locally of finite presentation. And then, yes, we use
[Tag01V8][1] If it makes you feel any better, his ultimate goal with this paper, and subsequent ones (which, incidentally, my thesis is a generalization of one of these papers) is to work in the same realm as the work of Harris-Taylor. In Harris-Taylor's seminal book/paper where they prove local Langlands for $\mathrm{GL}_n(F)$ they explicitly restrict only the schemes which are locally Noetherian (as does Kottwitz, if I recall correctly, in his original paper "On the points of some Shimura varieties over finite fields).
Katz-Mazur ---> Scholze: A smooth proper connected curve over a field is automatically projective. We may assume we're over $\overline{k}$. Let $X$ be a smooth proper conneced curve. Let $U$ be an affine open subscheme. Then, by taking a projectivization of $U$ (i.e. locally closed immerse $U$ into some $\mathbb{P}^n$ and take closure) and normalizations you can find an $X'$ which is smooth and projective containing $U$. Then, you get a birational map $X\dashrightarrow X'$. One can then use the valuative criterion to deduce this is an isomorphism.
An elliptic curve is connected. Note then that if $X/k$ is finite type, connected, and $X(k)\ne \varnothing$ then $X$ is automatically geometrically connected. Since any idempotents in $\mathcal{O}(X_{\overline{k}})$ must show up at some finite extension, it suffices to show that $X_L$ is connected for every finite extension $L/k$. Note that since $\Spec(L)\to \Spec(k)$ is flat and finite then same is true for $X_L\to X$, and thus $X_L\to X$ is clopen. Thus, if $C$ is a connected component of $X_L$ it's clopen (since $X_L$ is Noetherian) and thus its image under $X_L\to X$ is clopen, and thus all of $X$. Suppose that there exists another connected component $C'$ of $X_L$. Then, by what we just said the image of $C$ and $C'$ both contain any $x\in X(k)$. Note though that if $\pi:X_L\to X$ is our projection, then $\pi^{-1}(x)$ can be identified set theoretically as $\Spec(L\otimes_k k)=\Spec(L)$ and co consists of one point. This means that $C$ and $C'$, since they both hit $x$, have an intersection point. This is a contradiction. So an elliptic curve, being connected and having $E(k)\ne \varnothing$, is automatically geometrically connected.
Best Answer
Yes, 2 is a priori weaker than 1. And it is false in general.
There are examples of flag varieties specializing to smooth projective horospherical varieties; see Pasquier-Perrin
In particular, there exist a smooth projective morphism $X\to B$ with $B$ a smooth affine connected curve, and a closed point $b$ in $B$ such that $X_s \cong X_t$ for all $s,t\in B\setminus \{b\}$. However, $X_s \cong X_b$ if and only if $b=s$.
There are easier examples of this phenomenon, but this is the first one that came to mind.