Touch up paint is best for chips. Long scratches like that are just going to look like long scratches with a blob over the top of it if you try to paint it with a brush. Dealership paint or not, it isn't going to look very good.
Cleaner wax doesn't do a whole lot. It is just a solvent in a lotion and doesn't have much in the way of physical abrasives. Even with a machine you aren't going to get anywhere with a cleaner wax. It doesn't remove scratches.
If you can feel the scratch with your fingernail then it is too deep to polish out entirely. You may be able to lighten it up with polish, but it won't go away that way. You'll want to get something fairly heavy like Meguiar's ScratchX or GS27 if you can still find that. Rub it in the direction of the scratch and it should cut it enough to lighten it up as far as it will go. Then you'll need to use a lighter polish to remove the haze. There are a few steps to proper scratch removal, so don't expect this to come in one bottle with easy directions. If it was that easy to do, then nobody would have scratches on their paint.
One thing that I discovered last year is Dr. Colorchip. There are a couple posts about it on this forum even. That stuff is amazing for scratches and chips that can't be polished out. They send you body shop quality paint (meaning it is already thinned and is a perfect match) and some crazy solvent stuff. You rub the paint into the scratch and then wipe off the excess with the solvent. Somehow it leaves the paint in the scratch and removes everything else. I have been blown away by how clean the repair looks afterwards. Definitely an awesome fix. It isn't some temporary solution either. This is paint and not colored wax. It fills the scratch with paint and that is that. I would highly recommend it to anybody with a car.