The tune in Mudcat MIDIs seems to be based on a later version of "The Star-Spangled Banner." See (& hear) versions of it and of the first edition of "The Star-Spangled Banner" [for example, the first two notes are different]. All are from the Levy Collection.

Title: The Anacreontic Song.
Composer, Lyricist, Arranger: the Words by Ralph Tomlinson, Esqr., late President of that Society; [Music attributed to John Stafford Smith (1750-1836)].
Publication: London: A. Blands, Music Warehouse, 23, Oxford Street, 1784-1792]..
Form of Composition: strophic with SATB chorus
Instrumentation: voice, piano (two voices, bass and treble), German flute, guitar
First Line: To Anacreon in Heav'n where he sat in full Glee
First Line of Chorus: And besides I'll instruct you like me to intwine
Performer: as Sung at the Crown of Anchor Tavern in the Strand
Subject: Drinking
Subject: Singing
Subject: Gods--Greek
Subject: Myths
Call No.: Box: 000 Item: 001

Title: King and Country. An English Patriotic Song Taken from the Boston Patriotic Song of Adams and Liberty.
Composer, Lyricist, Arranger: Written by Thomas Paine A.M. To the Tune of Anacreon.
Publication: London: W. Milhouse, Musical Instrument Maker, 337 Oxford Street, n.d..
Form of Composition: strophic with refrain
Instrumentation: piano and voice
First Line: Ye Sons of Brittannia who bravely have fought
First Line of Chorus: As ne'er should the Sons of Brittannia be slaves
Subject: Patriotism
Subject: Liberty
Subject: Wars
Call No.: Box: 002 Item: 016

Title: The Battle of the Wabash A Patriotic Song.
Composer, Lyricist, Arranger: Written by Joseph Hutton, to the favourite Air of Anacreon in Heaven [following song is text for "Fort McHenry, or, The Star Spangled Banner."
Publication: Philadelphia: G.E. Blake, n.d..
Form of Composition: strophic
Instrumentation: voice and piano (two voices, treble and bass clefs)
First Line: In the dead of the night when aloud on the air
Performer: "Star Spangled Banner" Sung with great applause by Mr. Harding, at the Theatre Baltimore
Subject: Wars
Subject: Campaigns & battles
Subject: Flags
Subject: Patriotism

Title: The Defense of Fort McHenry (Star-Spangled Banner). The Grandest of American Patriotic Songs.
Composer, Lyricist, Arranger: The Immortal Lyric by Francis Scott Key. Music by James M. Stewart.
Publication: Washington, D.C.: E.F. Droop & Sons, 1894.
Form of Composition: strophic; different tune than "Anacreon in Heaven"
Instrumentation: voice and piano
First Line: O say can you see by the dawn's early light
Dedicatee: To Prof. Joseph H. Daniel of Washington, D.C.
Engraver, Lithographer, Artist: J.M. Armstrong Co., Music Typographers, 710 Sansom St., Phila.
Subject: Patriotism
Subject: Flags
Subject: Campaigns & battles
Call No.: Box: 000 Item: 005
Call No.: Box: 000 Item: 004

Title: The Star Spangled Banner. A Pariotic Song. [sic] [First edition].
Composer, Lyricist, Arranger: Adapd. & Arrd. by T.C.
Publication: Baltimore: Carrs Music Store, 36 Baltimore Street, n.d..
Form of Composition: strophic with chorus
Instrumentation: voice and piano; flute part included
First Line: O! say can you see by the dawn's early light
First Line of Chorus: O! say, does that star spangled banner yet wave
Engraver, Lithographer, Artist: (Tssb.)
Subject: Flags
Subject: Patriotism
Subject: Campaigns & battles
Call No.: Box: 000 Item: 000

"The Anacreontic Song" and a nineteenth-century version of "the Star-Spangled Banner" can be heard HERE. "The Anacreontick Song" from John Stafford Smith's Fifth Book of Canzonets (London, ca. 1785) [reproduced in Oscar Sonneck's Report on The Star-Spangled Banner, Hail Columbia, America & Yankee Doodle, 1909; rpt. Dover, 1972, p. 169] is more different; it is on The Hilliard Ensemble: The Singing Club (harmonia mundi [France] HMC 901153).
Even the 1884 version of "The Star-Spangled Banner" arranged by Collin Coe (CLICK HERE for MIDI) sounds somewhat different.

~Masato