Tolkien created many languages. Is there any list of all the languages he invented?
Ardalambion, a fansite dedicated to analyzing Tolkien's invented languages, has asked this very question; according to him the answer is somewhere between 2 and 20, depending on how permissive you are when defining a language (emphasis his):
The entire list given is:
Languages with real meat:
Some substance, but not usable languages:
Here, Ardalambion may or may not be counting Oromëan, Aulëan, and Melkian, the three known dialects of Valarin; assuming he does certainly makes the numbers more sensible.
This list evidently differs from the one discussed in the question; the discrepancy can be made up by incorporating "intermediate" forms of the Elvish languages, which Tolkien revised extensively on the way towards his "finished" Quenya and Sindarin, as well as some Mannish languages that never had a decent vocabulary or grammar, but just some rendered names. These include:
And there are a few others that are mentioned, but exist only in name:
The list in the book may also be including some languages Tolkien invented as a boy, or which weren't connected to the legendarium:
Which language is the "fourteenth" by the measure of the writer of that excerpt will depend on precisely what definition is used, which is not remotely clear. I have contacted the Tolkien Literary Estate directly, and will update when/if I get a reply.