Let $X$ be an algebraic variety and assume that there exists an $n\in\mathbb{N}$ and a closed immersion $\phi: X \hookrightarrow \mathbb{P}^n$. Now, is it a known and/or meaningful notion to define the degree of $X$ as the minimum of all degrees of such embeddings? In other words,

$\deg(X) := {\displaystyle \min_{ \substack{\phi: X \hookrightarrow \mathbb{P}^n \\\\ \text{closed immersion}}} \deg(\phi(X))}$

Another option I had in mind was taking $\deg(X)$ to be $\dim(X)!$ times the leading coefficient of the Hilbert Polynomial of its canonical ring. I would like to know if either notion is known, appears to have desirable properties and, possibly, if they are related.