There's a lot of crossover between products. When talking about wax, polish etc many over the counter products are both. For example Autoglym SRP is a polish, but it also contains a sealant/protection element that makes it an all in one, so gets used by many as a wax. You could generalise that anything with an abrasive in is a polish, but it's best to consider individual products and what they say they do.
A pure wax (as opposed to a cleaner wax or all in one) will have no abrasive content, will not remove any paint, and the product itself should not cause swirls. It will also not have any ability to restore paint that has scratches, swirls, or has oxidised. So it will need to be applied to good paint or you need to polish with another product first.
A machine polish will usually contain a diminishing abrasive that beaks down during the polishing process. As you work the polish, the abrasives get smaller and smaller, offering less cut and it will refine the finish as you work. A fine polish worked in this way will not cause swirls, it will remove them, but at the same time it will also remove paint. A coarse polish will also breakdown in the same way, but the coarser abrasives will often need finishing with a finer polish to remove any marring introduced by the coarser polish. Paint removal will also be faster.
A hand polish will generally have finer abrasives in, something like t-cut contains a non-diminishing abrasive, something like scratch-x contains a diminishing abrasive that needs to be worked thoroughly through it's diminishing stages.
So, a polish (in the context I have said above) will generally remove paint, can potentially introduce swirls if not used correctly, and should be used only when needed to restore the paint.
A wax (in the context I have used above) can be used often, as it won't remove paint, and provided your application technique and equipment is good, won't cause swirls or marring.
