I think the answer to my question is pretty simple, but I'm not able to figure it out.
The question is:
Find the distribution which corresponds to the moment generating function $\frac{2e^t}{3-e^t}$.
I've checked a few tables that list common MGFs and their associated distributions, but couldn't find any that matched this MGF. Do I need to modify the function before using a table?
Simply divide numerator and denominator by $3$ to obtain $$\frac{\frac{2}{3}e^t}{1-(1-\frac{2}{3})e^t}$$ Setting $p=\frac{2}{3}$ you have the MGF of the Geometric distribution $(1-p)^{k-1}p$, as in the third row of the table in https://en.wikipedia.org/wiki/Moment-generating_function#Examples