I believe there's two questions here: what is the earliest use of "aqua vitae" and when did it become synonymous with "distilled spirits". I'm going to answer the latter, when did "aqua vitae" become synonymous with "distilled spirits".
The French "eau de vie", Gaelic "uisce betha", Scandinavian "akvavit", and so on, all have their roots in the Latin "aqua vitae". The OED has the first recorded use in English as 1471.
1471 G. Ripley Compound of Alchymy in Ashm. 1652, 115 "With Aquavite ofttimes, both wash and drie."
My main source is The Quest For Aqua Vitae: The History and Chemistry of Alcohol From Antiquity to the Middle Ages by Seth Rasmussen.
While fermentation has been known since probably 6000 BCE, distillation of alcohol is much trickier. Low quality glass cannot stand the heat, and methods cooling the distillate vapor were inadequate. 13th century Italian improvements in glass making made distillation a much easier process, and higher and higher concentrations of alcohol were possible.
Earlier publications referred to "aqua ardens" (burning water) or "aqua flamens" (flaming water) referencing this thing which looked like water but which burned with a cold flame and gave a giddy sensation upon being drunk. Some thought this was Aristotle's "quintessence" or "aether", the 5th element that made up the heavens. This is also where we get "spirits", alcohol is the pure "spirit" of the wine. We're edging towards "water of life".
The earliest reference I can find to using "aqua vitae" to mean "very strong distilled alcohol" is in the late 13th century by several authors.
Taddeo Alderotti of Florence published De virtutibus aquae vitae as part of Consilia Medicinalia describing the distillation of wine in an alembic and aqua vitae as "of inestimable glory, the mother and mistress of all medicine" and that a little every morning "makes one happy, jocund, and glad". English physician Gilbertus Anglicus was recommending aqua vitae for travelers. And Arnaldus de Villa Nova who said of the name:
This name is remarkably suitable, since it is really a water of immortality. It prolongs life, clears away ill-humours, revives the heart, and maintains youth.
(European chemists have this tendency to gush about drugs.)
So there you have it. To the best of my knowledge, sometime around the late 13th century as the process of distillation became more refined and purer spirits were possible, the term "aqua vitae" became popular among European chemists.
The unrelated term "alcohol" did not come into use until the 16th century. This comes from the Greek "kohl" referring to a finely powdered form of antimony trisulphide used as eye makeup in antiquity. Arabic added their "al" prefix for "al-kohl". It then became a term for very fine powder. Then a fine part of anything. Then anything you pulverized or distilled.
In the mid-1500s, Paracelsus (Philippus Aureolus Theophrastus Bombastus von Hohenheim to his pals) is used "alkohol vini" or "the subtle part of wine" to refer to spirits distilled from wine. The "vini" (wine) part was dropped, and we have the modern "alkohol". Etymologically unrelated to "aqua vitae".