For your intended use, a dedicated camcorder sounds like a better choice. We picked up a Canon HD camcorder in 2009 before our daughter was born, and it's been great. Quick start-up, good in low light, and small and light enough to stick in a coat pocket (or wife's purse) without it being a nuisance.
What was close to $1k then will likely fall within your budget now.
The only minor complaint about the Canon was the relatively short life of battery that came with the camera. We picked up an extended-life battery for normal use, and just keep the original in the case for a backup.