Is the notation for '$a$ divides $b$' standard?

In number theory and algebra $\rm\:a\ |\ b\ $ in ring $\rm\:R\:$ means $\rm\: \exists\: c\in R:\ ac = b,\,$ i.e. $\rm\,ax=b\,$ has at least one solution $\rm\,x\,$ in $\,\rm R.\,$ Usually the ring is omitted, being understood from ambient context. This notation is used so widely that it surely can be considered standard notation.

Occasionally some authors introduce asymmetric variants, with the bar being tilted, or with "hooks" on the top/bottom of the bar, intended to give some visual cue as to which argument is to be pushed to the bottom of a fraction, or pulled to the top. However, as convenient as they may be, none of these asymmetric variants is in wide use.

That's right, the notation is absolutely standard, and reversing is simply wrong. So for example it is not true that $12\,|\,6$.

There is frequent student confusion about this, because $a/b$ is nowadays a common notation for the fraction $\dfrac{a}{b}$. That is a relatively recent development.