According to texdoc symbols: \mvert and \mid are identical and produce a relation. \vert is a synonym for | and both produce the same symbol, but should be used in the context of an ordinal, and should be used as an operator, not as a delimiter (p54, bottom). \divides once again produces the same symbol but should be used as a binary “divides” operator. \lvert and \rvert are left and right ...
As you can see, the top formula has \vert size as default. I want the latter formula, How do I change my command? I can't use the \left and \right command because there is no left or right in \vert. Any help is appreciated.
How can I change the size of \vert accordingly like \left and ... - TeX
math mode - Use \big\vert f (x) \big\vert or `\big\lvert f (x) \big ...
Also, you have $\vert$ and $\Vert$. Since you mention using a vertical bar as a binary relation, you may want $\mid$ which is kerned to sit midway between its neighbours. But that has no double-bar version so finally, you have the option of $\mathrel{\Vert}$ which is both double-bar and kerned to the centre.
In fact, I'd say that using \mid everywhere is as bad as (or maybe even worse than) typing | or \vert to denote each and every vertical bar. The macro \mid has a specific use to denote conditioning information.