Aesthetic trends do (in the end) turn out to be a movement, no?
The quotation you cited mentioned that it was not an organized movement. Some artistic "movements" are focused, with a clear leader or a certain group of individuals that set out on a specific path, or at least a group that perhaps share a common geographical location or education or political ideology that produced sense of common aesthetic purpose.
What that sentence means is that neoclassicism was not such an organized movement, but rather a trend that many different composers tended toward in very different ways. It was primarily a reaction against various excesses of late romanticism, expressionism, etc. - a return to relative aesthetic "order."
but what inspiration are we talking about here?
The Wikipedia article you cite lists all of this in the opening paragraph. Specifically, it says:
The neoclassical impulse found its expression in such features as the
use of pared-down performing forces, an emphasis on rhythm and on
contrapuntal texture, an updated or expanded tonal harmony, and a
concentration on absolute music as opposed to Romantic program music.
There are potentially other features (specifically, I'd mention use of older classical forms and structure for pieces/movements), but that's a pretty good summary. And the "updated or expanded tonal harmony" often meant a simultaneous imitation of some older features of classical or baroque harmony that was often expanded in a different way from how romantic music had expanded the harmonic palette.
That doesn't automatically make him a Neoclassical composer though
So what was it about the music that made it Neoclassical?
See the above quotation and the Wikipedia article you cited. Stravinsky in particular also drew on more classical ideas from ancient Greece too (as neoclassical elements in other arts, like sculpture, architecture, and poetry had during "neoclassical" phases).
Are we saying, therefore, in an attempt to understand precisely what
Neoclassicism is, that it is purely taking the works of others (18th
century) and putting your own unique style on it? To an extent, aren't
many classical composers always doing this?
One way of looking at it is that the neoclassical composers sometimes tried to compose as if the 19th century never happened. That is, what if we took Mozart's use of harmony, counterpoint, form, etc. and expanded it in a slightly different way from what romanticism did to the music? Critics of neoclassicism sometimes have said it might end up sounding like classical era music with "wrong notes." So, you return to some of the form and aesthetic of a much earlier era, but develop it in a different way compared to the way it played out in the 19th century.
Note: If you're looking for further clarification on what any of these terms mean concerning various features of neoclassicism, please modify the question to be more specific (as a lot of the features are already noted in the article that is cited in the question).