I would challenge you, Dante, my friend, to find some document from Jefferson (or Madison or some other Republican in those days) in which he calls himself a Democrat or Democratic-Republican.
Something from before the Progressive Era. At least before the Calvin Coolidge administration.
You have again used "R" for republican as in a party.
Washington denounced the Democratic societies and blamed them for unrest in the nation. Unrest like the Whiskey Rebellion
Dante has stated on numerous occasion that parties as we understand them did NOT exist during the years of Hamilton/Adams/Jefferson/Madison. The idea that they were truly the 'Democratic Party" "Republican Party" "Federalist Party" or any other organized political party is not factual in that sense. Followers of Jefferson as he stepped into the faction started by people like Madison and Beckely, started calling themselves democratic-republicans and later republicans. There is documentation on this.
We years later have to refer to them by something, so we use names to describe the factions and parties (not organized political parties as such) they belonged to and formed. These groupings formed the political parties of later years. We lazily use terms like First Party system, Second Party system...
Whether Jefferson or Madison referred to themselves as either a Democrat or Democratic-Republican was not the argument Dante made, but Dante says people who called themselves republicans have indeed left behind documentation where they refer to themselves as democratic-republicans.
Beard did NOT make up the phrase/label