Which is more correct, $\| \vec{a} \|$ or $|\vec{a}|$? The question is pretty self explanatory, but I’ve encountered situations where, for the length of some vector $\vec{a}$, to denote the length (or magnitude, which ever you prefer) as either $\| \vec{a}\|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ or $|\vec{a}|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ and I was wondering which notation is more widely accepted, per say? I’ve tried researching this and different websites actually use different notation. 

Any help is appreciated, thank you.
Definitions are not judged "correct" or "incorrect".  Both of your notations are used in various contexts in mathematics.  There is a problem only if the writer and the reader do not understand each other.
The first one is the more correct, in the sense that is preferable when you are dealing both with vectors and numbers and it is necessary to avoid misunderstanding.

The notation $|\cdot|$ indicates the absolute value for numbers, but it is also frequently and widely used for vectors when it is clear from the context.