Let $(M,\mathcal{O}_M)$ be a complex manifold. Let $\Delta\colon M\to M\times M$ be the diagonal map and $\mathcal{I}$ the kernel of $\Delta^{-1}\mathcal{O}_{M\times M}\to \mathcal{O}_M$. In this note, the author shows that
the sheaf of order $n$ principal parts
$$ \mathcal{P}^{(n)}:=\Delta^{-1}\mathcal{O}_{M\times M}/\mathcal{I}^{n+1} $$ are isomorphic to the sheaf of $n$-jets
$$ \mathcal{J}et^{(n)}:=(\mathcal{O}_M\otimes_{\mathbb{C}}\mathcal{O}_M)/\mathcal{J}^{n+1}, $$ where $\mathcal{J}$ is the kernel of the multiplication $\mathcal{O}_M\otimes_{\mathbb{C}}\mathcal{O}_M\to\mathcal{O}_M$ and both of them are locally free $\mathcal{O}_M$-modules.
(OK, it requires "if $M$ is smooth", but I think $M$ is smooth being viewed as a complex analytic space.)
Question 1: shouldn't $\mathcal{P}^{(n)}$, instead of $\mathcal{J}et^{(n)}$, be called the sheaf of jets as the usual notion of Jet bundle is constructed from $\mathcal{P}^{(n)}$?
Then, it follows that $$ \mathcal{I}/\mathcal{I}^2\cong\mathcal{J}/\mathcal{J}^2. $$ Here the first corresponds to the cotangent bundle hence is the sheaf of holomorphic differentials $\Omega_M^1$, while the later, by a result in commutative algebra seeing for example in Stack Project, is isomorphic to the sheaf of Kahler differentials $\Omega_{\mathcal{O}_M/\mathbb{C}}^1$.
However, from this MO question, Kahler differentials are very different from holomorphic differentials. We only have a surjective homomorphism $$ \Omega_{\mathcal{O}_M/\mathbb{C}}^1\longrightarrow\Omega_M^1 $$ which is NOT an isomorphism (consider $de^z-e^zdz$ for example). In particular, the $\Omega_M^1$ is locally free of rank $\dim(M)$ while $\Omega_{\mathcal{O}_M/\mathbb{C}}^1$ cannot be since the share the same dual module - the module of derivations.
Question 2: What's wrong in here?
(My guess: I misunderstand the above statements of isomorphic as the product is taken in a different category than the category of complex manifolds.)
Even ignore the isomorphism issue, the statement that $\mathcal{J}et^{(n)}$ is locally free is still strange for me since $\Omega_{\mathcal{O}_M/\mathbb{C}}^1$ is not locally free as mentioned before.
Question 3: What's wrong in here?