Congratulations from another Mom of twins!
My recommendation is to keep things VERY SIMPLE whatever you get - ie small, large LCD screen, easy settings, etc. You are going to be carrying around so much "stuff", adding a large, complex camera to the mix will just make it that much more difficult. I would search for something I could literally keep in my pocket, or a small pocket in your pocketbook, diaper bag, or whatever it is you'll be carrying (I actually stopped carrying a large pocketbook after my twins were born until I no longer needed the
enormous diaper bag; I just stuck my wallet and a few essentials in there along with everything but the kitchen sink, LOL.) Something you can pull out very quickly to get those great shots at the most difficult times. Buy it soon so you can get used to it before the babies come.
Groucho makes a good point and which could help simplify your choices. You'll mainly be taking pics of the kids, not landscapes for a while, so you probably don't need to worry about a huge optical zoom. There are lots of nice point and shoots which have a smaller optical zoom whereas the ones with larger zooms are more difficult to find. I agree with looking at the Canons and Fujis.
It's nice today there are such cameras available. My twins will be 10yo this year (where did the time go?) I had an auto 35mm film camera, and although I got plenty of great shots, I'm sure I could have gotten more - while paying less - had I had a digital camera. Here's one of my early favorites, the babies were about 5 weeks old and this was our XMas card. (Ignore the red eye on the dogs.)