As already mentioned in Dom's answer, everything you wrote is correct. I'd just like to add a few things concerning the term diatonic.
As you said, one meaning of "diatonic scale" is a scale with five whole tones and two half tones where the two half tones are maximally separated, i.e. at least by two whole tones. This includes the major scale and all its modes, and, consequently, also the natural minor scale.
However, this definition would exclude the melodic minor scale, which also has five whole tones and two half tones, but the two half tones are separated by only one whole tone. Also the harmonic minor scale would be non-diatonic according to this definition, because it has one augmented second. So there is also a more inclusive usage of the term diatonic scale, which includes these two other forms of the minor scale as diatonic. Since it seems that there's no generally accepted definition of diatonic scale (either including or excluding the melodic and harmonic minor scales), this is a source of confusion.
One nice property of diatonic scales, in the narrow, exclusive definition, is the fact that they can be generated by stacking perfect fifths on top of each other. E.g., if you take the sequence of perfect fifths
F-C-G-D-A-E-B you get all notes of the C major scale. If you take F as the root, you get F lydian. This property of the lydian scale, that it can be generated by stacking six perfect fifths on top of each other, was the main motivation for viewing the lydian scale as the fundamental scale in George Russell's Lydian Chromatic Concept.