It's more about context than it is about written music. It's called a third because it's the third step in the scale.
C major scale for example.
1 2 3 4 5 6 7
C D E F G A B
C major chord is
C E G: the first, third, and fifth steps (degrees) of the
C major scale.
It's the same case with minor triads. Here is the
C minor scale:
1 2 3 4 5 6 7
C D Eb F G Ab Bb
The C minor chord is
C Eb G: the first, third, and fifth degrees of the
C minor scale.
This is the simplest way to see it, but you can also see it as intervals, which are based on the distance between two notes.
1 minor second
2 major second
3 minor third
4 major third
5 perfect fourth
7 perfect fifth
8 minor sixth
9 major sixth
10 minor seventh
11 major seventh
One semitone is the distance between two adjacent keys in the piano. Your suggested notation is slightly incorrect (typically we count the semitones from the root). A major chord, in semitone notation, would be
r, 4, 7. A minor chord would be
r, 3, 7. Using the table above we can translate them:
r, 4, 7 =
root, major third, perfect fifth
r, 3, 7 =
root, minor third, perfect fifth
From this you can also notice that the second note of a chord is not always a major third. It can be a minor third, or anything else. The
C minor chord has a minor third, for example.
As you can see it's not a complicated stave of circles in lines, it's much simpler.
So, why use interval notation instead of your suggested notation? Because it takes into consideration the context, the scale, the key, which actually simplifies things a lot (even if you can't see it now). Your suggestion only uses the root as the reference, but nothing more.
Look at the
C major and
C minor scales again. In both cases the third note of the scale is the second note of the chord. Both are thirds (a major third in the major chords, and a minor third in the minor chord). This link is not present in your suggested notation, in which a major third would be a "5", and a minor third would be a "4".
That link is very important, because in tonal music everything gravitates around something (the key), everything functions in relation to something else. The interval notation implies these relationships, making things more clear and simpler. Once you start seeing things in terms of tonality, once you dive into music theory, this will become evident.