I hoped that someone besides me might have an answer for you, but I guess not...
First of all, understand that I am a video professional. I've been working as a television engineer for 30 years, and have also written, produced and directed some minor things. The things that I want from a camera, as someone with a LOT of background in television, may not be the things that you want.
Secondly, I'm not a parent. I can't speak for the needs of parents.
Digital still image cameras can take decent video. I shot quite a few videos with my still image camera and was pleased with how they came out. You probably know that I have some videos on YouTube -- Here's one that I shot with my still camera. For the best quality, click on the "HQ" box on the lower right of the screen:
http://www.youtube.com/watch?v=ci4pHK4tJok&feature=channel_page
The video is clean and the sound is clear... not bad at all. That was all one take, and notice that I didn't zoom in at all. Many still image cameras will not allow you to zoom when you are shooting video. Other still image cameras will allow you to zoom. Make sure you get a camera that will allow you to zoom if you want to be able to zoom.
I also have an HD video camcorder, and have loaded many videos from it onto YouTube. Here's one of a seagull on the shore of Lake Huron, in St. Ignace, MI. Like the previous video, click on the "HD" box on the lower right to get the best quality:
http://www.youtube.com/watch?v=SLt7zRCt_zo&feature=channel_page
Of course, the HD camcorder takes better videos but it cost a lot more money. I paid around $450 for the still image camera (a Canon PowerShot G7) and around $880 for the HD video camcorder (a Canon Vixia HF11). The video camcorder has come down a bunch in price and you can now get something like it for $600 or so.
There are still standard definition (SD) camcorders being sold, as well as a couple different types of high definition (HD) camcorders. If you get a video camcorder, I recommend a model that will allow you to record onto solid-state memory, such as SD memory cards. You will need a computer that's powerful enough to take the clips and process them, by stringing them together in an edited sequence or just be able to trim down a single clip.
I have a very good consumer digital camera and a very good consumer HD camcorder, and the following is true: Of the two, a digital camera will take the best photographs and a camcorder will take the best video. However, a digital camera will take decent video and a camcorder will take decent pictures. It all depends on what you'd like to do.
If you want the ability to do smooth zooms and to have a BIG zoom range, then you want a camcorder. It's not unusual for a consumer SD camcorder to have an optical zoom range of 40x. I've seen one consumer SD camcorder with an optical zoom of 50x! No digital camera will come close to that. HD camcorders have more modest zooms... the best I've seen there is 15x. Still, a camcorder will also allow you to do a s-m-o-o-t-h zoom, something that a still image camera can't do.
If you primarily take pictures and the occasional video is a nice thing to have, get a still image camera. You will get the best photographs from a still image camera, and that is your primary need. All I can say is that I'm having a blast with my HD camcorder, doing things I never previously thought about doing.
If you want HD, the best way to get it today is with a camcorder. There are just a couple of consumer still image cameras that can do true HD and they're both offbeat and expensive. Things might be different in a year or two (or five) when you begin to consider having a kid.
Anyway, that's my analysis. Let me know if you have any questions.
--Bob