Definition of sheaves of principal parts

Question

I am confused with the definition of sheaves of principal parts

As far as I understand, I have the following definitions.

1. [EGA IV. §16] Let $f:X \to S$ be a morphism of schemes and $\Delta: X \to X \times_S X$ the diagonal morphism, which is a locally closed embedding. We have short exact sequence over $X$: $$0 \to \mathcal{J} \to \Delta^{-1}\mathcal{O}_{X \times_S X} \to \mathcal{O}_X \to 0.$$ Then $\mathcal{P}^n_{X/S}$ is defined as $$\mathcal{P}^n_{X/S}:= (\Delta^{-1}\mathcal{O}_{X \times_S X})/ \mathcal{J}^{n+1}$$ The $\mathcal{O}_X$-bi-module structure of $\mathcal{P}_{X/S}^n$ is induced by the two projections $X \times_S X \to X$ (I omit the "obvious" details here). Besides, if $X/S$ is separated, then it's the same as $$ \Delta^{-1} \big(\mathcal{O}_{X \times_S X}/\mathcal{I}^{n+1}\big),$$ where $\mathcal{I}$ is the kernel of the surjection $\mathcal{O}_{X \times_S X} \to \Delta_\ast \mathcal{O}_X$

2. [Notes on Crystalline Cohomology. Chp 2] We have natural map $f^{-1} \mathcal{O}_S \to \mathcal{O}_X$ so that we can take the tensor product of $f^{-1} \mathcal{O}_S$-algebras: $\mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_S} \mathcal{O}_X$. Then we have $$0\to \mathscr{I} \to \mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_S} \mathcal{O}_X \to \mathcal{O}_X \to 0 $$ where $\mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_S} \mathcal{O}_X \to \mathcal{O}_X$ is the obvious multiplication map. Then we define $$\mathscr{P}_{X/S}^n :=\big( \mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_S} \mathcal{O}_X\big) / \mathscr{I}^{n+1}.$$ the bi-module structure is obvious from the tensor product.

Now my question is: what's the relation between $\mathscr{P}_{X/S}^n$ and $\mathcal{P}_{X/S}^n$ defined above?

I can check that there is a canonical map $$\varphi: \mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_S} \mathcal{O}_X \longrightarrow\Delta^{-1} \mathcal{O}_{X \times_S X} $$ from the universal property of tensor product. I think $\varphi$ could not be an isomorphism in general. For example, take any point $x \in X$, we have the stalks $$ \big(\mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_S} \mathcal{O}_X\big)_x = \mathcal{O}_{X,x} \otimes_{f^{-1}\mathcal{O}_{S, f(x)}} \mathcal{O}_{X,x} $$ which may not be a local ring but $$ \big(\Delta^{-1}\mathcal{O}_{X \times_S X}\big)_x = \mathcal{O}_{X \times_S X, \Delta(x)} $$ which is always a local ring. But i guess $\varphi$ may induce an isomorphism between $\mathscr{P}_{X/S}^n$ and $\mathcal{P}_{X/S}^n$.

Question: Is that true? If they were not an isomorphic, why do they give the same theory?