In modern discourse a Republic is a form of democracy. Pointing out that a Republic is not a democracy is in-group signaling among people who trade conservative talking points.

In 18th and 19th century discourse, meaning was more fluid and context dependent. Often writers were responding to criticism that any democratic system of government (of which the U.S. was undoubtedly one type) was intrinsically less stable and irrational than aristocratic or monarchical governments.

To quote John Adams,

A democracy, then, is a republic, as well as an aristocracy,
or any mixture of both.

The Federalist is a valuable work, and Mr. Madison’s part
in it as respectable as any other. But his distinction
between a republic and a democracy, cannot be justified.
A democracy is as really a republic as an oak is a tree, or
a temple a building. There are, in strictness of speech and
in the soundest technical language, democratical and
aristocratical republics, as well as an infinite variety of
mixtures of both.

(Letter to J. H. Tiffany, April 30, 1819)
(http://founders.archives.gov/documents/Adams/99-02-02-7104)

The John Birch society quotes "Samuel Adams" as saying,

Remember Democracy never lasts long. It soon wastes
exhausts and murders itself.

But the quote was actually from John Adams. (See http://founders.archives.gov/documents/Adams/99-02-02-6371) Apparently whoever threw that video together didn't actually read the real document. They probably just copied a quote (or in this case, a misquote) from a book of quotations, as people trying to make a point are so fond of doing these days.

If you read the actual letter the context of the quote selected by the John Birch society is clear. Adams is responding to what he believes is a mischaracterization of something he stated earlier. It's a scholarly argument among gentlemen about how intrinsically prone democracies are to violence.

I do not say that Democracy has been more pernicious, on the
whole, and in the long run, than Monarchy or Aristocracy.
Democracy has never been and never can be so durable as
Aristocracy or Monarchy. But while it lasts it is more
bloody than either.

Elsewhere in the letter it's clear the argument relates to some introspection by Americans on the bloody conflict in France. You can't assume that what Adams discussed in that letter he thought directly applicable to the U.S. The context was different and he wasn't making a grand argument about democracies generally, unless you believe he was happy to contradict himself five years later.

The notion that certain forms of democracy which check the passions of the majority are for that reason _alone_ intrinsically more stable is a very modern notion. It emphasizes one particular aspect of the structure of American government over all the others. The U.S. constitution employs several different devices. It was a hodge-podge of ideas. Few Founders expected the U.S. to last as long as it did. Remember, it was version 2.0. There's no reason to believe that they expected it to last much longer than version 1.0 (Articles of Confederation). Longevity was not the exclusive goal; it was at best an equal goal along with equality, security, prosperity, autonomy, etc. More likely longevity on the scale of human history was the lesser of those goals.