The fact that the name was "given by Verhulst" does not explain why it was given. Usually people pick names for a reason, but maybe Verhulst was an exception. Ok, I looked at the French wikipedia, which states "Le nom de courbe logistique leur a été donné par Verhulst sans que l'on sache exactement pourquoi." - "The name "logistic curve" was given to it by Verhulst, but no one knows exactly why". The reference rasch.org gives the following commentary:
"Verhulst writes "We will give the name logistic [logistique] to the curve" (1845 p.8). Though he does not explain this choice, there is a connection with the logarithmic basis of the function. Logarithm was coined by John Napier (1550-1617) from Greek logos (ratio, proportion, reckoning) and arithmos (number). Logistic comes from the Greek logistikos (computational). In the 1700's, logarithmic and logistic were synonymous. Since computation is needed to predict the supplies an army requires, logistics has come to be also used for the movement and supply of troops. So it appears the other meaning of "logistics" comes from the same logic as Verhulst terminology, but is independent (?). Verhulst paper is accessible; the definition is on page 8 (page 21 in the volume), and the picture is after the article (page 54 in the volume). "