Different browsers have different rules for which drivers they support. So you will get patchy support across different browsers if you have old drivers. Again, I highly recommend updating your drivers and this will likely enable WebGL for all browsers that support it. Unfortunately IE doesn't support WebGL, and Safari has it disabled by default.
Browser Quest probably drew all their artwork at double/triple/quad size with the point sampling baked-in to the images. You can do the same for your projects to get around the fact different browsers resize images differently.