Welcome to the forum.
First off, shooting sports is one of the most demanding things to ask of your camera. You need fast shutter speeds to freeze the fast moving action and to get the fast shutter speeds, it helps to have a large maximum aperture. You also need a long focal length because you have to be off of the field.
I'm not sure about the lenses that are built into the camera that you mention...but I'd guess that their maximum aperture (especially at the long end of the zoom) isn't very big. Also, you will usually find that the greater the zoom range, the lower the image quality.
Do they play their Lacrosse games in the daylight or under the lights in the evening? This makes a big difference because those lights don't really give you all that much light and getting fast shutter speeds is very hard.
There is something you can do, to get faster shutter speeds...and that is to turn up the ISO setting. However, the higher you go, the more noise you will get. That is part of those cameras you listed...they are not great at high ISO settings....mostly because they have rather small image sensors.
Another thing that should be considered is the shutter lag...how fast does the camera snap the photo after you press the button. I've found most 'digi-cams' to be painfully slow but the newer ones are better.
Basically, what I'm getting at is that any of those cameras isn't ideal for shooting sports. Of course, the idea set up for shooting sports would cost you a couple grand for the camera and several thousand for the lens.
You could try to find a happy medium or you could try to make the most out of a lesser camera.
My suggestion would be to look into an entry level DSLR camera. These cameras have much larger sensors that the digi-cams, so they have much better high-ISO performance. They also have a faster shutter lag. Most importantly, they use interchangeable lenses...so you are not limited to the lens that comes with the camera. You can buy a lens that is more suited to shooting sports. Of course, a DSLR camera is more expensive and the lenses range from $70 to $7000.
I know that said that you wanted to stay in that price range....and in good shooting conditions, those cameras might be adequate. But all too often people buy somthing like that, only to find that they are limited by it...and end up buying something better later.
