Pinto is a genetic trait that affects a horse's coloring. It's not specific to any breed or body type (in fact, most breeds carry pinto pattern genes in their bloodlines to some degree). There are several different pinto patterns out there (tobiano, sabino, splash, etc.), but they all share the common trait that they restrict the horse's 'base color' and leave areas of unpigmented skin and hair. There is a Pinto Horse Registry, but they are a color registry- not a breed registry. They allow any horse with pinto genes to be registered, and do not require any specific body type or traits other than the presence of one or more pinto pattern.
A Paint horse is any horse registered with the American Paint Horse Registry (APHA), whether it carries/expresses pinto genes or not. APHA is a breed registry, which highly favors horses with pinto genes & coloring, but also requires a specific body type. It was originally formed as a way to register Quarter Horses who the AQHA would not allow to be registered because they had 'too much white.' Part of the founding standard for the QH was that it be solid colored- but normal horse markings (like socks, blazes, etc.) are caused by pinto genes, and those genes have the capability of being expressed differently in each horse. Because DNA wasn't understood back then, those QH's who appeared to have inherited pinto patterns seemingly 'out of nowhere' were not allowed to be registered. They could, however, be registered with the APHA. The APHA has allowed in other horses since its original founding, but continues to be a stock-type horse with a preference for pinto coloring. APHA does not 'kick out' horses who don't have enough white. As a result, there are many solid colored Paint horses who either did not inherit a pinto pattern gene or do not express it very loudly. Those horses have a slightly different registration status, but are still considered Paint horses.