As a noun denoting a drink, cocktail is first recorded in The Morning Post and Gazetteer (London, England) of 20^th March 1798. In notes on the earliest mention of a drink called ‘cocktail’, I have:
– reproduced the text published in that newspaper;
– explained the terms used to denote the various drinks listed in that text;
– put forward a hypothesis as to the meaning and origin of the word cocktail as used in that text.

The second-earliest known uses of cocktail as a noun denoting a drink are from a purported diary published in The Farmer’s Cabinet (Amherst, New Hampshire, USA) of 28^th April 1803.

This diary does not define the precise meaning of the word cocktail. The only clues are that the putative diarist:
– is “a lounger”, who first describes himself, on waking up, as feeling “queer”, having “no appetite” and feeling “rather squally”;
– then remarks, on drinking “a glass of cocktail”, that it is “excellent for the head”.

All that can be inferred, therefore, is that the drink called cocktail is apparently a liquor that the alleged author seems to take as hair of the dog.

(As a noun denoting a drink, cocktail was defined three years later—cf. 1806: earliest definition of ‘cocktail’ (mixed drink with a spirit base).)

This is the text published in The Farmer’s Cabinet (Amherst, New Hampshire, USA) of 28^th April 1803—this newspaper was published by Joseph Cushing:

Amusing Miscellany.
For the Farmer’s Cabinet.

Mr. Cushing,
The enclosed “brief chronicle,” of a short period of a lounger’s life, was lately found on the plain, lost probably from the pocket of the careless author. From the careless dress in which the loose minutes appear, it is doubtful whether they were intended for public inspection; it is more probable that they were written either to avoid the attacks of the author’s old and inveterate enemy—Time, or perhaps to excite a smile on the countenance of some beloved fair one. The author would probably be totally unable, at this distant period, to collect a history of the events he has here recorded, from the dark cells of treacherous memory, since but for this little memorial they had already been covered by the broad mantle of oblivion. We request you to publish them for the gratification of your readers, because we conceive them to be a picture of existing manners, correctly, though perhaps unintentionally, drawn. Besides, if the author still frequents the haunts where he has been accustomed to lounge, though he never looks into an old paper, still perhaps his eye may chance to glance on the article thus rescued from oblivion, if conspicuously displayed in your Miscellany.

JOURNAL.

FRIDAY.—Waked at 7 by the bell—wonder what people mean by disturbing one so early after an Assembly: turn’d and doz’d ’till 9: got up, and dressed—felt queer; took a cup of coffee—no appetite.—10. Lounged to the Doctor’s—found Peter—talked of the girls—smoked half a cigar—felt rather squally: Van Hogan came in—quiz’d [note 1] me for looking dull—great bore.—11. Drank a glass of cocktail—excellent for the head; all sauntered away to see the girls: Miss —— not up: Went to the Squire’s—girls just done breakfast. Mem. [note 2] Girls not so bright after dancing. Talked of the weather—then of the walking—then of the weather again—was very witty—Peter not quite so brilliant. Went to the Col’s. found the girls very lively and sociable—drank a glass of wine—talk’d about Indians—call’d Miss —— a Squaw—all laugh’d—damn’d good one—talked about the walking—insisted that the more muddy it was, the better walking—all look’d queer: nothing else to say—jogg’d off.  Call’d at the Doct’s. found Burnham—he looked very wise—drank another glass of cocktail. All went to the Printing Office—began to smoke—Mons. —— look’d into an old paper—stupid fellow—never look into papers myself—spoils the imagination.—1. Strol’d [?] home—dinner ready—ate a little. Mr. —— fatigu’d me with politics—don’t like it—so threw myself into an easy chair—fortunately got into an easy posture and smoked a cigar.—3. Went into the Doct’s.—found Burnham and Van Hogan—drank a little gin bim  [note 3]—vile stuff—all went down to the Squire’s—got into spirits—talk’d very bright—introduced the weather again. Mem. Always to talk about the weather,—can always say witty things about it: Took a walk with the girls—felt tired—returned—saw the Court House open, so strolled in—heard lawyers wrangling and disputing—hate disputations—happened to laugh heartily at something Burnham said; an odd, musty looking fellow, with a huge white staff call’d out Silence! insolent fellow—all mere boors—no regard to gentlemen—walked [?] off. Smoked another cigar at the Doct’s.—N. B. Doctor’s a famous lounge. Tried to walk up the turnpike—too fatiguing—went into Peter’s Office—he very busy—quiz’d him—swore business was a bore; bright one—all laugh’d. Thought I should’nt say any thing better,—so went home to tea.—7. Stroll’d into Atty’s Hall—call’d for cards—play’d till 12—got rather hazy. N. B. Bad wine, never drink such again—went home, to go to bed—wonder what makes me feel sick—folks all asleep—went to bed very tired.

NOTES

1 Here, the verb quiz is used in its original sense, that is: to make fun of, to mock, to tease—cf. origin of ‘quiz’ (“Vir bonus est quis?”)?.

2 The word mem. is an abbreviation of Latin memorandum (est), meaning (it is) to be remembered.

3 I have not discovered the meaning of bim in gin bim.