What was the first documented mention of American English different from British English?

Few languages are created by a dictionary writer.^*

The differences emerging between 'English English' and 'American English' were observable a lot earlier than the very desire for American Independence from Britain came into being, let alone to fruition

We therefore need to go further back than the publication of that dictionary.

A certain Mr Webster just followed suit of the trend to make identity politics with language when in 1790 he called the English spoken in the United States "the American language", which in 1789 he referred to as the American tongue:

The body of the people, governed by habit, will still retain their respective peculiarities of speaking; and for want of schools and proper books, fall into many inaccuracies, which, incorporating with the language of the state where they live, may imperceptibly corrupt the national language. Nothing but the establishment of schools and some uniformity in the use of books, can annihilate differences in speaking and preserve the purity of the American tongue.

Webster: Dissertations On The English Language, &C. 1789 https://www.gutenberg.org/files/45738/45738-h/45738-h.htm

An intersting contemporaray to Jefferson would be a certain Mr Witherspoon, whi is credited with coining the word "Amercanism":

… the Scottish churchman John Witherspoon, a signer of the Declaration of Independence and member of Congress, as well as president of the College of New Jersey (renamed Princeton University in 1896) from 1768 until his death in 1794, was one of America’s important political figures and intellectuals awkwardly caught in the cross fire of the Anglo-American battle of the languages.

Witherspoon understood and appreciated Jefferson’s celebration of neologisms and other types of vocabulary expansion as natural parts of language development, but he had no taste for the extreme forms of language he heard cropping up in all walks of life in the country.

He deplored American slang and indiscriminate, undisciplined looseness of expression on the part of the better educated, including members of Congress, lawyers, and clergymen: “vulgarisms,” “common [grammatical] blunders arising from ignorance,” “cant phrases,” “personal blunders,” and “tautology.” “I have heard in this country,” he wrote in 1781, “in the senate, at the bar, and from the pulpit, and see daily in dissertations from the press, errors in grammar, improprieties and vulgarisms which hardly any person of the same class in point of rank and literature would have fallen into in Great Britain.”

Among the Americanisms that he said he heard everywhere—he claimed he was the first to use that term to describe differences between British and American English—were the following: the use of every instead of every one, contrive it for carry it, mad for angry, …

Before the rebellion of the colonies became an official war, we read:

As perception of the country as a nation separate from England grew, so too did perception of language differences. In January 1774, and anonymous writer (possibly John Adams) issued a proposal in the Royal American Magazine for a national academy:

The English language has been greatly improved in Britain within a century, but its highest perfection, with every other branch of human knowledge, is perhaps reserved for this land of light and freedom. As the people through this extensive country will speak English, their advantages for polishing their language will be great, and vastly superior to what the people of England ever enjoyed.

"…at the hands of Americans": The American (English) Language", Linguistic Nationalism http://helmod.vtcath.org/america.html

Those who still know how to properly spell aluminium were among the first to notice that American grew into something different.

When Thomas Jefferson's book came to England, people noticed some peculiarities:

Jefferson came in for some English criticism of his use of Americanisms in his only book, Notes on Virginia, in 1787. His use of the word belittle (a perfectly good word today, of course) in it inspired this piece of mockery in the European Magazine and London Review:

Belittle!—What an expression!—It may be an elegant one in Virginia, and even perfectly intelligible; but for our part, all we can do is, to guess at its meaning.—For shame, Mr. Jefferson! Why, after trampling upon the honour of our country, and representing it as little better than a land of barbarism—why, we say, perpetually trample also upon the very grammar of our language? … Freely, good sir, will we forgive all your attacks, impotent as they are illiberal, upon our national character; but for the future, spare—O spare, we beseech you, our mother-tongue!7

As proscriptive guardians of 'proper language tick, calling a naturally evolving and diverting language to be 'full of errors', the dictionary writer Samuel Johnson observed in 1756:

In 1756, the year after he published his famous dictionary, he coined the term “American dialect” to mean “a tract [trace] of corruption to which every language widely diffused must always be exposed.” He had in mind an undisciplined and barbarous uncouthness of speech. With typical hyperbole on the subject of Americans, he once remarked, “I am will- ing to love all mankind, except an American … rascals—robbers—pirates.”

— Peter Martin: "The Dictionary Wars: The American Fight Over the English Language", Princeton University Press, 2019.

This difference in language evolution paths was apparently beginning much earlier:

By 1720, the English colonists began to notice that their language was quite different from that spoken in their Motherland.

The first “official” reference to the “American dialect” was made in 1756 by Samuel Johnson a year after he published his Dictionary of the English Language. Johnson’s coinage of the term “American dialect” was not meant to simply explain the differences, but rather, was intended as an insult.

Years earlier, however; as early as 1735, the English were calling our language “barbarous,” and referred to our “Americanisms” as barbarisms.

Kathy Weiser: "Evolution of American English", Legends of America, May 2017.

Freely available starting points to read up on this:

H. L. Mencken: "The American Language The First Differentiation" https://en.wikisource.org/wiki/The_American_Language/Chapter_7

Peter Martin: "Noah Webster’s civil war of words over American English" https://aeon.co/ideas/noah-websters-civil-war-of-words-over-american-english

Peter Martin: "Americans and Brits Have Been Fighting Over the English Language for Centuries. Here’s How It Started" https://time.com/5604227/american-british-english-literature/

_{^* Hat tip of course to Marc Okrand as one rare example.}

John Pickering's A vocabulary, or collection of words and phrases which have been supposed to be peculiar to the United States of America of 1816, does not use the phrase "American English"; but he refers (p.10) to "a language, that is to be called at some future day the American tongue!". This latter is contained in "An Essay on the Present State of the English Language in the United States".

So, while it does not meet the letter of your question, I think it is a good candidate for answering the spirit.

I thought at first that there was one instance of "American English" in the Vocabulary, but it turns out that this is in a separate paper bound in to the volume, a review by of Johnson's dictionary, from the American Quarterly Review, Vol IV, No VII, but I haven't found the date of that. A handwritten note says that this is by Pickering. The context is about the likelihood that "the several nations in Europe, ... because their proximity to England ... will oblige them to study the European and not the American English".

(It was The Prodigal Tongue, as in my comment above, that directed me to Pickering)