In textbooks on algebraic number theory and $p$-adic numbers, I quite often find two different statements and they are all called Hensel's lemma or at least different versions of Hensel's lemma. The statements I have in mind are as follows:
Version 1: Let $f \in \mathbb{Z}_p[X]$ and $a_0 \in \mathbb{Z}_p$. Assume that $$|f(a_0)|_p < |f'(a_0)|_p^2.$$ Then there is a unique $a \in \mathbb{Z}_p$ with $f(a)=0$ and $$|a-a_0|_p \leq \bigg|\frac{f(a_0)}{f'(a_0)}\biggl|_p.$$
Version 2: Let $k$ be the residue field of $\mathbb{Z}_p$ and for $f \in \mathbb{Z}_p[X]$, let $\overline{f}$ be the reduction of $f$ to $k$. Consider a monic $f \in \mathbb{Z}_p[X]$. Assume there are two polynomials $g_0,h_0 \in k[X]$ which are coprime over $k$ and with $\overline{f}=g_0h_0$. Then there are two unique polynomials $g,h \in \mathbb{Z}_p$ with $\overline{g}=g_0$, $\overline{h}=h_0$, $(g,h)=\mathbb{Z}_p[X]$ and $f=gh$ in $\mathbb{Z}_p[X]$.
I am wondering what the logical relation between those two versions is.
Question: What is the exact logical relation between version 1 and version 2? Does one implies the other? Are they equivalent?
A related question is here: Deducing Hensel's lemma for polynomials. The answer seems to suggest that version 2 might not imply version 1 (although a definite proof is not provided as I understand it).
I found these lecture notes: https://math.mit.edu/classes/18.785/2017fa/LectureNotes9.pdf. On page 7, the author claims that version 2 does imply version 1 but he doesn't give a proof.