I'm wondering this too, however I'm starting to get the impression that dine is for plural words, and din/dit are singular. Din is for -n words and dit is for -t words. (Or so I've read, I'm learning just as much as you are :p)
Way late but I'm guessing that it's because "Dog's" is counted as possessive (e.g. "The dog's bone") rather than a contraction of "dog is".