My 2 Cent's Worth...
I don't have a toddler, but I take pics of our cats who are our current children and they move fast too (when they're awake, that is). Two speed tricks I found that I didn't see mentioned in this thread that should work with whatever camera you end up with:
1- use high-speed memory cards for a tiny bit of advantage
2- turn off the review/display that holds the shot for you to see what you got after each shot. That way you can snap several shots in a few seconds time and improve your chances of capturing the cuteness. You can go back later to look and cull.