I have the same stripper tool and it works flawlessly, I don't want to insult you but are using it correctly? Mine does have an image of which way to place the coax into the tool before you spin it to cut and trim. My compression tool is a little different model but I found that you really need to make sure you apply some pressure to hold the cable from backing out when the compression is actually being done.
All the foil and braid are electrically the same. Are you doing F-connectors because sometimes they can be a bugger to push on there far enough. If the white dielectric isn't almost butted up to the opening at the other end you probably don't have it pushed on far enough.
that little stripper tool should have a couple of blades in that will will leave everything correct for you after its used, I am away from home at present but if you can wait until the weekend I can probably make you a little video that will show you