Side remark: Huygens had previously used that name for the graph of logarithm, in Discours de la cause de la pesanteur (1690, p. 169):
... there was a curved line, which I had examined much before, which was of great use in this research. One can call it Logarithmic or Logistic, for I don’t see that anyone has yet named it, although others also considered it before.
Latin version with elaboration by G. Grandi in Opera reliqua (1728, pp. 130, 135, 149–288). Same in the Cyclopaedia (1728: Logistic Line, Logistic Spiral, Logistical Arithmetic) and in Roger Boscovich, De cycloide et logistica (1745, p. 80) and also Theoria philosophiae naturalis (1763, pp. 260–261). Lalande, already quoted by Mauro Allegranza, confirms in his Vol. 3 (1771, nº 3915):
Logistics is the name one used to call algebra; later one gave it to the logarithmic curve; today it is devoted to this little kind of logarithms [“logistic logarithms”]; perhaps this word comes from λογὶζομαι Colligo, because algebra packs many things in few characters.
As to “algebra”, Cajori says (1896, p. 26): “Greek mathematicians were in the habit of discriminating between the science of numbers and the art of computation. The former they called arithmetica, the latter logistica.” Later Vieta used logistica numerosa and speciosa, and maybe not coincidentally, a Belgian biography of Stevin had recapped, shortly before Verhulst’s paper (Goethals 1841, p. 13):
Logistica numerosa was properly arithmetic; logistica speciosa was algebra, which since Vieta, contemporary of Stevin, split into two branches (...). The logistic due to Monk Barlaam, who flourished in 1350, was originally the arithmetic of sexagesimal fractions, more curious than useful. This word was long kept; and by the end of Stevin’s career, it was still used by the mathematician Beyer who published, in 1619, a treatise on decimal and sexagesimal logistic.
So the word had been through quite a few different meanings when Verhulst (1845) co-opted it out of obsolescence again — with no apparent explanation. Whence understandable confusions, such as Wikipedia linking Boscovich to the wrong graph.