Notation/name for the number of times a number can be exactly divided by $2$ (or a prime $p$)

If you want to be excessively fancy, you can call it the $2$-adic valuation $\nu_2(n)$.

Number theorists use the notaton $\operatorname{ord}_p(n) = k$ to mean that $p^k$ is the largest power of $p$ that divides $n$. In other words, $p^k | n$, but $p^{k+1}$ does not. It is also of note that $\operatorname{ord}_p(0) = \infty$ for each $p$.

See here for more.

You might call it the "highest power of $2$ dividing $n$," but I'm not aware of any snazzy standalone term for such a thing. However, I have seen it notated succinctly as $2^k\|n$, which means $2^k$ divides into $n$ but $2^{k+1}$ does not.