The first variant is dependent on a series of unsubstantiated theories. When mammals starve, they release a hormone that tells the body to burn through the last reserves of fat. This has been collected from starving humans and injected into fat humans and has resulted in weight loss. Rabbit and human hormones are very similar (enough that the urine of a pregnant human female can be injected into the nape of a female rabbit's neck, and the human pregnancy hormone will trigger ovary growth in the rabbit as if the rabbit were pregnant - this is the old rabbit pregnancy test). If the starvation hormones are also compatible between rabbits and humans, then it is possible that eating a starving rabbit could cause you to loose more fat that you would get from the rabbit.
The second variant is based on the simple reality that if you are so starved that there is nothing but rabbit to eat and there is no greenery to supplement your diet with, chances are the rabbit is also starving (they don't have secret food stashes). A starving mammal of any variety is not going to have the fat content of a healthy mammal, and rabbits are no exception. To anticipate otherwise just doesn't make sense.
So now to completely bust the rest of that myth. From our friends at the USDA:
U.S.D.A. Statistical Breakdown of Various Meats
Rabbit 20.8 10.2527.9795
Chicken 20.0 11.0 67.6810
Veal 18.8 14.0 66.0910
(md. - fat)
Turkey 20.1 22.2 58.31190
(md. - fat)
Beef 16.3 28.0 55.0 1440
Lamb 15.7 27.7 55.81420
(md. - fat)
Pork 11.9 45.0 42.02050(md. - fat)
So as anybody can see rabbits have virtually the same amount of fat and calories as a chicken. The differences are minimal. The cholesterol level is somewhat less than chicken, but still not dramatically lower (difference is about 25%):
And none of this is taking into account the sources of fat on a rabbit outside the meat. Rabbits also have strips of pure fat that run from the posterior end of the kidneys up to the diaphragm. Then of course there are the kidneys, heart, and liver (and the lungs, brain, and skin if you are really wanting that last gram of fat) which also have some fat in them.
So, baring a nasty winter, with nothing but snow and ice, after a famine, and with no other sources of nutrients, YOU WILL NOT BE LACKING FAT FROM EATING RABBIT MEAT!!!