Normalization and blowup are different operations

Given: A field $k$ of characteristic $0$ (or different from $2$), the plane ${\mathbb{A}}_k^2$, the cuspidal cubic $C=V(y^2-x^3)$ with singular point the origin $0$, its normalization $\nu \text{\hspace{0.17em}⁣}:\tilde{C}\to C$, and the blowup $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_0{\mathbb{A}}^2\to {\mathbb{A}}^2$.

Choice. The Axiom of Choice is assumed as inherited from the normalization and blowup constructions used by the cited items. (The Axiom of Choice).

Normalization of a reduced curve is finite: Every reduced curve of finite type over $k$ has a finite normalization morphism $\nu \text{\hspace{0.17em}⁣}:\tilde{C}\to C$, unique up to unique isomorphism over $C$.

First blowup of the cusp y^2=x^3: Blowing up the origin of the cusp $y^2-x^3$, in the chart with $y=xs$ the total transform is $x^2(s^2-x)$, so the strict transform is the parabola $s^2=x$, which is regular and meets $E=(x=0)$ at the single point $s=0$; the other chart contributes no further intersection.

Blowing up a rational point of a smooth surface: The blowup of a smooth surface at a $k$-rational point is smooth with exceptional curve ${\mathbb{P}}_k^1$; over an affine neighbourhood the blowup is the incidence subscheme $V(xv-yu)\subseteq U\times {\mathbb{P}}_k^1$ with charts $\mathrm{Spec}R[y\mathrm{/}x]$ and $\mathrm{Spec}R[x\mathrm{/}y]$, an isomorphism away from the center.

Finite morphisms of schemes: A finite morphism has finite fibres; a morphism whose fibre over a point is positive-dimensional is not finite.

A proper quasi-finite morphism is finite: A proper quasi-finite morphism of schemes is finite.

Normalization is unchanged under finite birational maps of reduced curves: A finite birational morphism of reduced curves induces an isomorphism of their normalizations.

Birational morphisms of integral finite-type schemes: For integral finite-type schemes over $k$, birationality means that the generic point maps to the generic point and the induced function-field map is an isomorphism. An isomorphism over a nonempty dense open gives these properties.

embedding dimension and regular local ring: A Noetherian local ring is regular when its embedding dimension ${\dim }_k(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)$ equals its dimension.

Finite-variable polynomial algebras over fields are integrally closed: A polynomial algebra over a field is integrally closed; its localizations are integrally closed as well, so ${\mathbb{A}}_k^2$ is normal and its normalization is the identity.

Blowing up a point on a singular surface need not be smooth: For the curve $y^3-x^5$ the first chart of the point blowup is $k[x,s]\mathrm{/}(s^3-x^2)$, whose local ring at the origin is not regular, so that point blowup does not normalize the curve.

Blowups of finite type ideals are locally H-projective, and proper and Strict transform of a closed subscheme: The blowup is proper over its base. The strict transform is a closed subscheme of the scheme-theoretic inverse image of the curve.