The United States is not a direct democracy, in the sense of a country in which laws (and other government decisions) are made predominantly by majority vote. Some lawmaking is done this way, on the state and local levels, but it’s only a tiny fraction of all lawmaking. But we are a representative democracy, which is a form of democracy.
And indeed the American form of government has been called a “democracy” by leading American statesmen and legal commentators from the Framing on. It’s true that some Framing-era commentators made arguments that distinguished “democracy” and “republic”; see, for instance, The Federalist (No. 10), though even that first draws the distinction between “pure democracy” and a “republic,” only later just saying “democracy.” But even in that era, “representative democracy” was understood as a form of democracy, alongside “pure democracy”: John Adams used the term “representative democracy” in 1794; so did Noah Webster in 1785; so did St. George Tucker in his 1803 edition of Blackstone; so did Thomas Jefferson in 1815. Tucker’s Blackstone likewise uses “democracy” to describe a representative democracy, even when the qualifier “representative” is omitted.
Likewise, James Wilson, one of the main drafters of the Constitution and one of the first Supreme Court Justices, defended the Constitution in 1787 by speaking of the three forms of government being the “monarchical, aristocratical, and democratical,” and said that in a democracy the sovereign power is “inherent in the people, and is either exercised by themselves or by their representatives.” And Chief Justice John Marshall — who helped lead the fight in the 1788 Virginia Convention for ratifying the U.S. Constitution — likewise defended the Constitution in that convention by describing it as implementing “democracy” (as opposed to “despotism”), and without the need to even add the qualifier “representative.”
To be sure, in addition to being a representative democracy, the United States is also a constitutional democracy, in which courts restrain in some measure the democratic will. And the United States is therefore also a constitutional republic. Indeed, the United States might be labeled a constitutional federal representative democracy. But where one word is used, with all the oversimplification that this necessary entails, “democracy” and “republic” both work. Indeed, since direct democracy — again, a government in which all or most laws are made by direct popular vote — would be impractical given the number and complexity of laws that pretty much any state or national government is expected to enact, it’s unsurprising that the qualifier “representative” would often be omitted. Practically speaking, representative democracy is the only democracy that’s around at any state or national level.
Now one can certainly argue that some aspects of U.S. government should become less direct, and filtered through more layers of representation. One can argue, for instance, that the 17th Amendment should be repealed, and that U.S. senators should no longer be elected directly by the people, but should return to being elected by state legislators who are elected by the people. Or one can argue for repealing state- and local-level initiative and referendum schemes. Or one can argue for making the Electoral College into a deliberative body, in which the electors are supposed to discuss the candidates and make various political deals, rather than being elected solely to vote for particular candidates. And of course one can equally argue for making some aspects of U.S. government more direct, for instance by shifting to truly direct election of the president, or by institute a federal-level initiative and referendum.
But there is no basis for saying that the United States is somehow “not a democracy, but a republic.” “Democracy” and “republic” aren’t just words that a speaker can arbitrarily define to mean something (e.g., defining democracy as “a form of government in which all laws are made directly by the people”). They are terms that have been given meaning by English speakers more broadly. And both today and in the Framing era, “democracy” has been generally understood to include representative democracy as well as direct democracy.