Which is better: canon, nikon or fuji?

We're looking at getting a good camera to take pictures of our kids. We're looking at a budget of 200-350 and we've found a few that look good. We were told we had to get a canon, regardless of megapixles and zoom. I have a hard time believing that but what do I know. Which is the better camera and why?
Canon powershot xs130 (12MP, 12x zoom)
Canon powershot xs230 (12MP, 14x zoom)
Fujifilm S4000 (14MP, 30x zoom)
Nikon L120 (14MP, 21x zoom)

We want good quality, fast (to catch kids) and we don't want to pay for a name… We want to pay for the best camera.

You can compare the four cameras side-by-side here:

http://www.dpreview.com/...kon_cpl120

One thing you might want to consider is the battery type. The Canon SX230 uses rechargeable lithium-ion batteries while the other three use AA batteries. I know of some cameras that just eat up AA batteries for no apparent reason, so I actually prefer lithium-ion batteries. If you have the extra money to spend on quality batteries from time to time though, AA batteries might be a better option for you.
The Canon SX230 is also the most compact and lightest, which can come in handy if you travel a lot and want pocket it or keep it in a bag/purse.
Those are just some things to consider, but be sure to look at all the specs in the link above!

There isn't a best camera. There's what fits you. You hands are going round it.
Try some. Then decide. Canon fans choose Canon. Nikon fans choose Nikon.
Photographers choose by what fits their needs at the time and how well it fits with any other other equipment they have, not by a name.
As for specs and quality and stuff if you have to ask then all of those cameras are far better than you can make full use of.
These days there are very few bad cameras and there's easy access to overload cameras.
For a lot of people it's like buying a car the wrong way round
Buy a Ferrari and then ask on here how to drive a car.
Any camera now selling for $100 and upwards will give you sharp clear pics of the children growing up including fast action if you set the ISO high enough or if it's auto ISO and the light is good.
For $200 you've got more choices than a cat in a fish market
Any of them will get you great pics.
What fits you best though so you can actually use it you can only find out by handling them.
Go to a photo shop and try some.
When you've bought one read the name on it if it's of much interest.
Of my 30 -odd old cameras and 6 modern ones I don't give a cuckoo's call sign which one I use for most jobs.
No-one else will be able to tell anyway whether I used a Nikon, Canon, Minolta, Fuji, Pentacon, or my 40 year old trusty old Exactas with pin-sharp Zeiss lenses and a Weston light meter.

Nikon Coolpix L120 Digital Camera with 14.1 Megapixels, 21x Optical Zoom, 3in LCD Display, HD (720p) Movies, 5 Way Image Stabilization System.

It also supports macro shooting at as close as just 1cm to the subject. Its image sensor shift method and electronic vibration reduction will help for blur-free photo shooting.users can take out snapshot sized prints as well as larger prints of A4 size