In the study of local theory for holomorphic varieties, the Local Parametrization Theorem states that in $\mathbb{C}^n$,for any irreducible germ of holomorphic variety $V$ at 0, there exist a nonsingular linear change of coordinate under which the natural projection $\pi : \mathbb{C}^d \times \mathbb{C}^{n-d} \to \mathbb{C}^d$ is a *finite branched holomorphic covering of $\mathbb{C}^d$*.

On the other hand, the Noether normalization Theorem, applied in the context of reduced holomorphic algebra, states that for every reduced holomorphic algebra $A$ there exist an integer $d$ and a finite injective algebra homomorphism $\phi : _d\mathcal{O}_0 \to A$.

It seems to me that the two theorems are telling us the exact same thing, but one is stated in a more geometric language while the other in a algebraic one. Is this true?

If so, why do we have two separate theorems? Were they developed independently?

If they are not equivalent. How can I see the difference?