Now I have a better idea. Don't worry about your English - it's a lot better than some people who use it as their first and only language.
Regarding zoom, disregard the 'x' factor..it's only a relative term which has no meaning as what your picture will look like. For example, a camera with a zoom focal length of 36 to 180mm has a zoom factor of 5x. A camera with a focal length of 28 to 140mm also has a zoom factor of 5x. Go to this site and see the difference of the same scene at different focal lengths:
Now, if you will be taking pics of children inside, a focal length of 28mm (wide angle) is desirable. If you intend to take pictures of wildlife outside, a longer focal length, say 400mm or more is preferred. If you buy a camera that has a focal length of 28 to 400mm, you have a zoom range of roughly 14.3. If you think, therefore that a lens with a zoom range of 20x is better than one of 10x, stop - it ain't necessarily so. The longer the range, the more prone to defects in the lens. It costs a lot of money to design and build a high zoom lens...a $200 18x zoom is liable to fall quite short of your expectations.
Regarding low light pictures, some brands/models are better than others. If you want to see what the cameras on your list can do, go here:
Plug in two of the cameras on your list and compare the "still-life" images at ISO 800 (that's a speed setting you'll probably need to use in low light situatioins). How do they look? Is that what you were hoping to see? Compare each of the cameras on your list this way. Once you have your favorite selected, plug in the Fuji S100fs and compare. What do you think?
Regarding the ability to capture an image of a child before he turns his head, that can be difficult with a point & shoot camera. If you select one with a short lag time, and you pre-focus your shots, you stand a chance. As far as continuous shooting, or burst modes, all of the cams you selected will do that, but some arer better than others.
If you plan on taking a lot of indoor pics, you might want to consider a camera with a hot shoe, so that you can add an external flash.
Hope that helped somewhat.