You need to understand the difference between close and open voicings. In close voicings the notes are arranged in thirds or seconds. Inversions of close voicings are created by moving the lowest note up by one octave. So the close voicings of C7 are
C E G Bb (root position)
E G Bb C (first inversion)
G Bb C E (second inversion)
Bb C E G (third inversion)
Note that on the guitar most of these voicings are (close to) unplayable because they would require very wide stretches.
The type of voicings shown in the video are called drop-2 voicings. They are very easy to play on the guitar, and they also sound good. To create a drop-2 voicing, you take any of the above voicings and drop the second note from the top by one octave. So the first voicing above
C E G Bb becomes
G C E Bg (by dropping the second note from the top, the G, to the bass). In this way you can create all the voicings shown in the video.
As pointed out in another answer, for open voicings (and drop-2 voicings are open voicings), it is the bass note that determines the name of the inversion. So for a C7 chord, the note E in the bass means that it is the first inversion, regardless of the arrangement of the other notes in the chord.
Another popular type of voicing on the guitar is the drop-3 voicing, where the third note from the top is dropped. So, e.g., the close voicing
C E G Bb becomes
E C G Bb (that's the one you expected to see as the first inversion!). Unlike drop-2 voicings, drop-3 voicings are usually not played on four adjacent strings, but the bass note is played on either the low E or on the A string, and then one string is skipped, and the remaining three notes are played on the next three adjacent strings. So the voicing
E C G Bb would be played as (from low to high)
12 X 10 12 12 X
X 7 X 5 8 7