A Paint is a Paint whether it is registered or not. Just like an unregistered Quarter Horse is still a Quarter Horse. A Paint is a horse whose parents are also Paints(or QH or TB, since the APHA allows that) Because they are an outcropping of the QH, they tend to have a stock horse like build. Since they allow TB parentage, there can certainly be a taller, leaner looking horse and it would still be considered a Paint.
A pinto is a horse with one of the pinto markings/genes as described by Verona. It can be of any parentage.
The word pinto describes both the colour and the gene.
The word Paint describes a breed.