As the other answer says, the word anacrusis for a preliminary unaccented syllable seems to have been invented by Johann Gottfried Jakob Hermann. Besides backing water, there may be another possible place where Hermann could have gotten the name from.
Looking in a French dictionary at the etymology of anacrouse, they pointed me to the Pythian games—contests held in Delphi every four years, honoring the God Apollo.
There was a very specific musical contest in these games.
I then found a description of the relevant contest in the Pythian games in a translation of Strabo's Geography. This is the only place I can find where this contest was described, and thus if the contest is where Hermann got the name, he probably found it in Strabo.
Strabo was a Greek geographer and historian who lived in the first century BCE. The following sentences (which are nearly the only relevant material) are an excerpt from Strabo:
There was anciently a contest held at Delphi, of players on the cithara, who executed a pan in honour of the god.
The players on the cithara were accompanied by players on the flute, and by citharists, who performed without singing. They performed a strain (Melos), called the Pythian mood (Nomos). It consisted of five parts; the anacrusis, the ampeira, cataceleusmus, iambics and dactyls, and pipes.
So the ancient greeks indeed seem to have used ἀνάκρουσις for the first movement of a musical composition, or at least for this specific musical composition in the Pythian games. Although this is somewhat different than the meaning Johann Hermann (see the other answer) assigned it in 1817, one can see why Johann Hermann might have chosen anacrusis for this meaning.
Why was anacrusis used for the first movement of the composition in the Pythian games, and did this have anything to do with its use in sea battles? I don't know. But certainly the idea that starting a song with a few unaccented syllables is like pushing a ship back from the docks seems to be an invented etymology.