""C E G", "E G C", and "G C E" are all inversions of a C triad, but not "C G E", because of the gap between the C and the G, where an E would fit"

"In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and (in counterpoint) inverted voices. The concept of inversion also plays a role in musical set theory."

and

"Third inversions exist only for chords of four or more tones, such as 7th [& 9th etc] chords. In a third-inversion chord, the 7th of the chord is in the bass position. For example, a C major 7th chord in third inversion consists of B in the bass position, with C, E and G above it— being intervals of a 2nd, 4th and 6th above the (inverted) bass of B, respectively."

There are also "Inverted intervals" - which can slightly negate Piers' pure point about "C G E" not being an inversion of a C major chord. Interestingly too, in full score Orchestral Score Music Notation (and [Big] Band Score too!), the "C G E" of a C major chord can be shared across different instruments in different octaves to get particular timbres... :-)

There are also further concepts such as "Inversional equivalency &
Musical set theory" and "Counterpoint" related.... :-)