The main problem you have is that what you are filming is the most challenging environment for both a video camera and its operator. For the night-time events you have extremes of bright light (because of the floodlights near the starting line) and great darkness (also near the start, at the base of the cars, and when the cars are further down the track). Then you have a subject which travels very fast, potentially starting away from the camera operator, coming very close as it goes by, then getting far away again – all of which makes focusing and exposure a nightmare!
Conventionally, the best type of videocamera for high contrast scenes and low light situations is actually a video-capable DSLR (stills) camera. However I would warn you in this case to stay away from them. DSLRs are very hard to operate quickly and easily in “live action” situations such as your car races; their lenses are also very, very difficult to focus on a moving subject at the best of times so trying to keep an immensely fast car in focus would be tricky.
So your best option, I think, will be a camcorder despite its (generally) poorer performance in low light. I think you will have to accept that although footage should be great in daylight and acceptable on the start line, you will inevitably get a grainier, noisier image as the cars move away than you would with a DSLR. Whether this image noise bothers you is a matter of personal taste.
Of the three camcorders you mention, the Sony NX30 is little more than a consumer handycam dressed up in a slightly more professional body. That does not necessarily make it a bad camcorder but for reasons I will come to in a moment, it may not be the preferred choice.
Incidentally, one of the features of the NX30 (and others, including the XA20) is connections for professional audio (“XLR” connectors). I am unsure whether you intend to do interviews and how important audio is to your filming. If it is irrelevant, and if you decided that the NX30 is sufficent for your needs, then you could save money by buying the consumer version of this unit instead of paying the extra for it being labelled “NX30” and categorised as part of Sony’s professional line-up. Just a thought.
The NX70 is a camera with a specific reason to its existence – to be dust and water-resistant. Obviously this could be useful when filming outside (any other camcorder would need a cover put over it if it starts to rain, which makes operation much more awkward). It is a perfectly decent camera but if you have no need for water-resistance then it is not necessarily the right choice for you. If it does rain often when you film then this camera could be perfect as rain-covers really are a nuisance.
The XA20 is designed to be a camcorder that is easy to use under live action scenarios yet complex enough to permit sophisticated operation. You can put it into fully automatic mode and it will do a decent job but any camcorder will struggle to auto-focus on such fast-moving targets and in such dark conditions so you are going to have to practice and train someone to get used to focusing manually. You might be able to put the camera onto auto exposure (I recommend “TV Priority” with a shutter speed of 1/60th if you’re in the USA) and it will do its best to set the exposure correctly. On the other hand, it may not be able to cope with your conditions and you may need to set exposure manually as well. Without actually trying it, you won’t know if it will cope.
As for manual exposure, this is where things get more complicated.
You see, the downside of all these small cameras is that they have only one control ring (usually for focus) with perhaps a small dial for adjusting exposure manually – and small dials are not ergonomic. Trying to adjust them quickly and without wobbling your shot is very hard. That’s why “pro” cameras have three full-size control rings: one for focus, one for zoom (often little used as the rocker switch on the side of the camcorder is better) and one for exposure.
So, depending on how much effort you wish to put into the filming, it should yield better results if you are happy to buy and use a camcorder with more controls. They tend to cost more, alas.
The only inexpensive one I know of is the Panasonic AG-AC90. A lot of people might be put off this because it has tiny, tiny sensors to pick up the image (and therefore shouldn’t be any good in dark situations) however I have seen a lot of reviews which say it’s actually quite good. The other benefit of small sensors is that it’s much easier to focus (contrast this with bigger sensors = shallower depth of field = more trouble keeping things in focus hence the reasons DSLRs are awkward to use on moving subjects). It has good image stabilisation too which helps when you’re handheld rather than on a tripod.
Another option could be the new Sony HXR-NX3. It’s at the top end of your budget but it apparently has excellent low-light performance from the tests I’ve seen. One substantial downside is that it loses autofocus very badly when panned quickly (as you would when filming a car coming past) so you’d have to use manual focus if you got this one.
I hope all this hasn’t made the decision even worse! Yours is a very tricky problem to solve which depends on a) how exacting you are about what quality of footage you want, b) how much you will pay and c) how much effort you are willing to do to become competent at using the camera.
I don’t have a “best” solution for you but would strongly recommend, if at all possible, that you borrow (or rent) a handful of these different cameras and try them out both in advance of a race and then at the racetrack as well. Only then will you see which one suits you best.