Hey there my namesake.
I am personally biased towards the Sony : http://www.avsforum.com/t/1429201/the-official-sony-pj710-pj760-pj780-pj790-owners-thread
I have last years PJ710
But objectively if you do get it just be aware that it is not the magic be all that will totally eliminate shakes.
You have to train your arm and your way of moving to get the best shot. No floppy arms !
The lens will float and keep moving slightly in some instances if you use too much zoom.
When you are zoomed in
at max this effect gets worse if you yourself also start to move. (I find all cameras do this not just the Sony.)
But it is amazing for going up and down stairs. Also great when in a vehicle on a bumpy road. See the test vids I posted above.
I tried walking on sand and that was a disappointment- part of it was me- I kept looking at the LCD and not watching where I was going so the lens had to compensate for my exaggerated missteps.
Haven't tried running- but the Sony's promo vids. sure look good. (For my future experiment, I will be picking up an external stabilizer mount for it- when I figure which one is the best for max portability/performance.)
I was also worried about the AVCHD format- it turned to be the best format to use with my chosen NLE. No issues at all.
And yes since you have a young one- the built in projector will be awesome- my brother just got a lower model Sony PJ since he liked mine so much and his three younger daughters love that instant feature.
You can even use it to have a big look at what you have just shot and then decide to retake the shot if you are able to. Much better than the small LCD. Provided of course you have a smooth surface to project onto.
Speaking of which- the still pics are really good even though I only have jpg on my model. I never bothered to replace my point and shoot cameras I always use this instead.
That Carl Zeiss lens actually works goodly for my amateur needs.
Oh yeah- iAuto works really well- set it and forget it.
From long shots to macros it switches really fast and seamlessly and chooses the best settings. (The only times I see it hesitating and flick through multiple settings is when I move from an extremely bright sunny environment to a really dark one. But that's to be expected- I have learned to compensate and choose the best angles to eliminate this even when on the move. A learning experience for me.)
I played around with the manual zoom and other manual settings- but it's not my thing so never really did an in depth test of those settings.
[Edited- to correct info highlighted in bold. Originally had the word "out" which is incorrect.]