For any given triad there are six possible arrangements of the notes.
For example: CEG, CGE, GEC, GCE, ECG, EGC
Why are only three of them taken account of, ie. root, first and second inversions?
Theorists have decided that there will only be three 'inversions' - root, with base note (C) at the bottom, 1st inversion, with E at the bottom, and 2nd inversion, with G at the bottom. Obviously using C major as the key and chord.
So, C E G and C G E are both root position triads. E G C and E C G both 1st inversions, and G C E and G E C both 2nd inversions.
You'll see that some are close voicings, where the three notes are as close as possible to each other, while the others are open voicings - which could have other chord tones in between the played notes.
Even with larger chords, such as played on guitar or piano, where there may be two or three of the same name notes, but in different octaves, the inversion is named from the lowest note. You may be interested to find out about drop chords too.
It follows that chords with four notes will also have a third inversion, and so on. But voicings will have many combinations at and after that many notes.
If you're trying to take everything into account, there are more than six arrangements - C4-E4-G4 is not the same as C4-E4-G5.
Any language has different levels of specificity. If you tell me you bought a car, that's more general than if you tell me you bought a blue car, and that's still more general than telling me you bought a blue 2017 Focus.
Music is the same. "C" as a chord name is general - any arrangement of CEG will fit the description. "First inversion C" or 6 with an E in figured bass is more specific... but not as specific as standard notation, which will tell you the exact arrangement.
We really don't need a tool more specific than an inversion, but less specific than standard notation, which will tell you if it's G4 or G5 in the voicing.