Not a stupid question at all! But, yes, only the bass note is taken into account when naming the inversion of a chord. The voicing above this is unimportant. Indeed, the bass note may be doubled, as can any other chord-tones. (Although, this may be inappropriate, if following the rules of specific styles of harmony or counterpoint.) Also, notes above the bass note can appear in any octave, so the spacing is also not considered when naming the inversion of a chord.
So, yes, E-C-G is first inversion; G-E-C is second inversion; C-G-E is root position.
There are three commonly used types of notation for denoting the inversion of a chord:
- lower-case letters following the roman-numeral (which denotes which scale degree is the chord's root). For example, in C Major: Ia would be a C Major chord in root position; Ib would be a C Major chord in first inversion; Ic would be a C major chord in second inversion.
- figured-bass shows the intervals to be used above a given bass note. So, in effect, this also gives information about the chord's inversion. A root position chord requires a 5th and 3rd to be added above the bass note, so is termed a 5-3 chord (although in practice these are considered to be implicit, so no figures are required for a root position chord. For the same reasons: 6-3 denotes a first inversion (the 3rd is implicit, so often only 6 is used); 6-4 denotes a second inversion.
- in popular music, slash chords show both a chord name and the pitch of a bass note which must sound as the lowest pitch in the chord. This bass note may or may not be a chord tone of the chord written as the first part of the slash chord. For instance: a C/G chord is a second inversion C Major chord, as the note G is part of a C Major chord; a C/D chord is also a valid slash chord, but this is not an inversion of a C Major chord.
It should be noted that all the chords and methods of notation described above, have thus far dealt with triads. Inversions of seventh chords are also common. In this case, the root, 3rd and 5th in the bass are still described as root position, first inversion and second inversion respectively, but if the 7th is in the bass this is a third inversion chord. (In the case of seventh chords, different figured bass numbers are required - see article here for figurings of inversions of seventh chords.)
Arguably, the figured-bass system best shows that only the bass note affects the naming of chord inversions, and not the spacing or voicing. Due to octave equivalence any G in a root position C Major chord is denoted by the figure "5", even if it actually sounds a twelfth or two-octaves-and-a-fifth (or more) above the bass note. The same is true for the use of 6, 4, and 3 (and indeed 7 and 2 in seventh chords); they need not be exactly a 6th, 4th, 3rd, 7th or 2nd above the bass note, but must be a note with the same "letter-name", even if this is a compound interval (which can be described as: an interval of less than an octave, plus one or more octaves).