Recently I have seen random variable distributions described in two ways:
$$ X \sim Nb(r,p) \\ X \stackrel{d}{=} Nb(r,p) $$
Both indicating that $X$ is a negative binomial random variable with $r$ successes and $p$ probability of failure. Do these mean the same things? Are there any other ways of writing this?
Yes.
Both symbols are commonly used to mean "has a distribution of".
I've encountered "$\sim$" much more often than "$\mathop{=}^d$", possibly because it is more convenient to typeset, but your mileage may vary.
Negative binomial is typically indicated by both-capitalised "NB", or the whole word "Pascal".
I cannot recall any other variations, but it is quite possible they exist. A good text book should have a glossary of symbols used.
$$X\sim\mathrm{N\!B}(r,p)$$