The “democrats” (as in democracy) and “republican” (as in a republic) are not very descriptive identifiers for a democratic party in a republic (though I guess you could argue a two-party system isn’t very democratic).
In many other countries, parties are labelled as “Progressive”, “Conservative”, “Socialist”, “Christian Democratic”, “Social Democratic”, etc. that give you some idea of what they stand for / believe in, albeit in some cases, they can be misnomers and be things they want you to think they believe in (e.g. the various “democratic” parties in one-party states). Why are the American parties so boringly named with such broad descriptors?
p.s. Whatever happened to the Bull Moose party?
Just double checked online, it looks like you and schnuritto are both partially correct. The “Democratic Republicans” did splinter off to the Democrats and the Republicans, but these were different from the modern kind and they seem to be unrelated to each other. This was the “National Republican” party that supported John Adams and were conservative too, but not related to the “Republican” party of today. These national republicans later merged into the Whig party, pretty interesting I think.