Yes, there was never a party called "the democratic-republican party" there was just the old republican party, which split to form the democratic party and the republican party we know today. But that's confusing to people, so historians teach it as having existed. However, they also tend to generally say that the actual name was the republican party. They're trying to emphasize that the democratic party wasn't some splinter group that barely accounted for many people in the party. The split was catastrophic, with neither side retaining all of its original ideals.