I have recently been learning about sheaves of differentials, and in particular the conormal sheaf. That is, the sheaf of differentials corresponding to the diagonal morphism. I have become rather confused about the difference between normal, conormal, tangent, and cotangent in algebraic geometry.
Let $X$ be a scheme. For the sake of this question, everything will be local so say $X = \text{spec} A$ is affine. If $Y$ is a closed subscheme with closed immersion $i: Y \longrightarrow X$, let $\mathcal{I}$ be the corresponding sheaf of ideals. I recently learned the definition of the conormal sheaf, $\mathcal{I}/\mathcal{I}^{2}$, on $X$, or the pullback of this on $Y$. I wanted to relate this to the idea of a tangent space at a point. Suppose $p\in X$ is a closed point and give it the reduced induced subscheme structure. In other words, we assign the residue field $\kappa(p)$ to the point $p$ making $Z = \{ p \}$ a scheme. Then as usual the closed immersion $j: Z \longrightarrow X$ corresponds to a surjection of sheaves, $$ j^{\#}: \mathcal{O}_{X} \longrightarrow j_{*}\mathcal{O}_{Z}. $$ The sheaf $j_{*}\mathcal{O}_{Z}$ is then the skyscraper sheaf on $X$ assigning the residue field $\kappa(p)$ to the point $p \in X$. We obtain an exact sequence of stalks at $p$, $$ 0 \longrightarrow I_{p} \simeq \mathfrak{m}_{p} \longrightarrow \mathcal{O}_{X, p} \longrightarrow \kappa(p) \longrightarrow 0 $$ But I am also familiar with the notion of the Zariski cotangent space, which is given by $\mathfrak{m}_{p} / \mathfrak{m}_{p}^{2}$. But in this case, it seems to be telling me that the Zariski cotangent space is precisely the same as the conormal space, which seems wrong. Have I completely misunderstood what these things are, or is there some flaw in my reasoning above? Is this simply expressing the fact that for a closed point, the conormal space is isomorphic to the cotangent space (in some kind of perfect pairing)?