I believe the Jeffersonians were called simply the Republican party. It was during Andrew Jackson’s time that the Republicans split into the Jacksonians (which became the Democratic-Republicans and then the Democratic party) and the National-Republicans headed by John Quincy Adams.
Yes, but it’s not uncommon for later historians to refer to Jefferson’s party as the “Democrat-Republicans”; it’s not strictly authentic, but it’s clear.
Jefferson called his party the “Republicans”, but his opponents called them the “Democrats” as a term of abuse. Eventually, they adopted the word as a badge of honor, and became today’s Democrats, today’s Republicans being a party formed from the wrecks of half a dozen failed parties in 1856. To avoid confusion, it is not uncommon to call the early now-Democrats-then-Republicans “Democrat-Republicans”, no matter what they called themselves then, just as it is conventional today to refer to dates between January 1 and March 24 in years prior to 1752 as, e.g., February 11, 1731/32, though people didn’t at the time.
We get added confusion in that democrat (small d) and republican (small r) are both commonly used political terms, independent of the current U.S. political parties. I think we’d save lots of confusion if the political parties would change their names to something fairly unique and reasonably recognizable, like Whigs and Tories. Or even Bull-moosers.