The opening words of the Platonic dialogue Timaeus are:
εἷς, δύο, τρεῖς: ὁ δὲ δὴ τέταρτος ἡμῖν, ὦ φίλε Τίμαιε, ποῦ τῶν χθὲς μὲν δαιτυμόνων, τὰ νῦν δὲ ἑστιατόρων;
One, two, three, — but where, my dear Timaeus, is the fourth of our guests of yesterday, and our hosts of today?
clearly showing that the word εἷς "one" was used for counting in the same way as all other positive integers.
If at some point later on (some) Greek mathematicians accorded a special status to the number $1$, I would be tempted to compare this to a modern discussion of whether $0$ is or is not to be included in the natural numbers -- or whether the zero ring is an integral domain or not. It is a technicality, of the sort that only becomes meaningful once one has run into the necessity of "constructing" objects that have already been understood on a more intuitive level.
To be sure, there seems to be a certain attraction to the statement as presented by Ian Stewart: it is intriguing to consider the possibility that the Greeks somehow thought "differently" about numbers than we do. It is a statement that borders on the unprovable. Heidegger is even more outrageous, writing somewhere (in "Modern Science, Metaphysics, and Mathematics", which is reprinted in the "Basic Writings" collection) that the Greeks didn't even consider 2 to be a number: since 1 is the unit, and 2 is only ever a "pair", and not a collection of "two" separate objects. Only when the number three is first encountered, does the concept of number really start to shine forth -- or so the the little magician from Messkirch would have it. It sounds good, but I have yet to be convinced that there is more to it than idle speculation.