"rabbit starvation" refers to a diet of mostly lean protein, deficient in fat. So much so that it results in malnutrition.

Eating nothing but rabbits is one way to get it, but is not really about "subsisting on small herbivores". It's the fact that the meat is very lean, not fatty. Apparently "mal de caribou" is the same thing, and Caribou / Reindeer are not small.