Has anyone developed a "cheat sheet" of sorts that describes the appropriate use of distribution types for different types of data? For example, beta for coin-type data (e.g. Therapy versus control), poisson for counts...
Check the Statistics 110: Probability course by Joe Blitzstein (Harvard University). Materials and lectures are freely available online as well as his handbook Introduction to Probability. Blitzstein provides many examples of common probability distribution and "stories" behind them, that make it easier to memorize what is the general idea behind them.
You can check also the paper
Lawrence, M. and McQueston, J.T. (2008). Univariate Distribution Relationships. American Statistician, 62(1): 45–53.