Yes, they did
(that is, the quoted assertion is not true)
Some members of this party indeed called themselves "Democratic-Republicans", although not consistently.
The Wikipedia article for this party states (in its main text) what I was taught in grade school, namely that the party was called "Republican" during the period it was in existence. Yet further along, they walk it back:
The name Democratic-Republican was used by contemporaries only occasionally.
And the footnote supporting that assertion has this:
After 1802, some local organizations slowly began merging "Democratic" into their own name and became known as the "Democratic Republicans"
along with a number of examples, including a link to an 1802 letter to Jefferson which says:
"At a large and respectable Democratic Republicans Meeting Held in Dover at the House of Daniel Cooke on the 24TH of March 1802, agreeing to the Notice given -- Major Abraham Pierce was called to the Chair, and Doct. John Hansen was appointed secretary."
(apologies for my attempts to transcribe the handwriting)
Even better, an 1803 campaign broadsheet, "To the Independent Republican Citizens of the County of Philadelphia." (reproduced at the Library of Congress) has:
Friends and fellow citizens!
You are this day presented with a strange and novel spectacle, two distinct tickets by two separate parties both styling themselves Democratic Republicans.