Is there a term for the distribution of numbers-of-digits of a bounded uniform distribution
As @YuvalFilmus suggests, these distributions are very close to being geometric. They aren't geometric, since they're bounded, which is what threw me off, but think about it this way:

A geometric distribution corresponds to the number of Bernouli trials up to and including a first success (in some formulations: not including the first success). The distribution parameter p is the probability of succeeding in each individual experiment. Now,


Our first "experiment" involves sampling the most significant bit (or byte): Will it be 0, or something else? A 0 is a success, something else is a failure.
Our second "experiment" is the same, but for the second-most-significant bit (or byte).
and so on.


In a geometric distribution proper, experiments never cease until the first success. In our case, $k$ experiments are conducted, and if you've failed them all then you're simply at 0, with probability $2^{-k}$.