Another vote for a non-cheap DSLR. Nikon DX40 serves me well, no noticeable lag. Point-and-shoot pocket camera won’t cut it.
You’ll notice the lag if you step up. Your camera has around 100ms shutter lag. Better than a point and shoot, but it’s there. Mine has about 37ms shutter lag, and you’ll never notice it unless you use a camera that has about half of that, and they exist. I did, and now I kinda hate my camera. Well not really, but it was humbling.