Mathilde uP - The best cameras concerning lag/focus currently availabe in the prosumer class are:
Additionally, please consider the SD-9 Sigma. While it is a SLR, it has recently plummeted in price to about $1000-1100 USD. A little more than you suggested you wish to spend, but a good camers none the less with fast AF. It doe shave poor low light peformance though, concerning noise.