A compact camera that does video, or a compact video camera that does still photography?

My ideal is to have one machine that does both video and still photography. I want it to be small enough to fit in my pocket. I want it to have good optical zoom, maybe wide angle too, HD, large solid state memory, high quality imaging, 8 or more megapixels for still photos. Image stabilization. Good low light capabilities.

Which way should I go? Should I look at cameras that do video? My daughter has one, but the zoom sucks and the video is impossible at a distance (no stabilization). Or should I be looking for a video camera that can take stills? Do they all take stills these days?

Can you suggest any models that might meet my needs? I’d like it to be under $300, if at all possible.