I use my DSLR for short family videos all the time, but it's a higher than your budget even used or refurbed. It works for me since I already take my camera anyway, and it already shoots video, so no need for an extra device.
For a video camera GoPro seems to be the standard these days.
Can't really help much there as I am not familiar with them. I don't know specs, but is would look for one that has high ISO settings which will be like "night vision" since it can pick up more light in darker areas.
When I went to Disney, I used ISO 16,000 - 25,600 regularly especially inside the rides and my shots and videos came out very nicely. I don't know if any GoPros will go that high on ISO though.
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