I want a camera with no lag time so that I can capture my children. My fine motor skills are really clumsy and I don't want to be faffing about trying to half press. I just want to point and press...and press and press away!
You can press any dSLR or any other camera you wish without pressing halfway first. As to the quality of the resulting picture, that is another story. You just completely miss the point. There are ways to speed up the process of taking the picture. LiveView is a major contributor to shutter lag. Disable it and use the optical viewfinder instead. Focusing is the other major event that takes up shutter lag. Learn how to eliminate focusing using depth of field in manual focus mode. If your subject is relatively stationary, you only have to press half-way once momentarily for the camera to get its readings. You can then shoot rapidly after that. Set your shutter to continuous shooting even for more machine gun pleasure. Of course you can only do all of these if the lighting is optimum. If you are shooting in low light then you have no chance at all to catch your kids.
Get yourself a Leica M9 or any manual film camera. Then you can click, click... and click away! Of course if you wanted pictures that were in focus, you'd naturally want to take a more patient approach and make sure you focus the image first. Similarily, even on a DSLR camera with autofocus, you'll want to half-press, make sure the subject you want to be in focus is in focus, compose, then shoot. Good luck.
Any DSLR with a continuous AF, which most should have, you probably want one with more AF points for more chance of getting the shot, something like a Canon XSi or T2i would be a good enough starting point, but use in P mode with AiFocus selected. Will probably work best if you pre-select your desired focus point and track this over your subject. If you want the AF to do more for you then you need something like the 7D which has more points in selectable zones, and each of the more points are more sensitive than on the budget bodies. I would also advocate a fast prime lens such as the 50mm f1.8 or a fast aperture constant zoom like the Sigma 17-50 f2.8, to give the AF system as much light to work with as possible.
Any SLR camera can do this, there is no requirement that you go through the half-press step. The camera will meter and focus as quickly as the scene allows, and take the shot. Depending on light levels, that may be near instantly, or it may still take some time to get it right. The fastest way to use the camera will be with manual exposure (metering doesn't matter) and fixed manual focus. Manual focus does you no good if you can't control it fast enough, or if you can't really see your subjects. As a compromise solution, manual exposure and auto focus will be the quickest way to work. Live View is only a pain in the posterior if you are using a Canon or Nikon body. Live View adds no delay when using Sony or Olympus bodies with that feature, due to the way it is implemented using a secondary sensor to drive the live view system. Pentax, I don't know for sure, but I believe it works like the Canon and Nikon versions. Even the low end bodies will allow you to shoot at 3 frames per second continuous, providing there is enough light. The machine gun approach is frowned upon in most cases, but it may work for you. You can shoot a burst, then select the 1 or 2 images you like best out of the burst. Just use a large memory card.