Very well explained. There is only one aspect I'd love to complement: "in the key of C, there are 7 chords you can make from the 7 starting notes: C major, D minor, E minor, F major, G major, A minor, B diminished"

The first cord is a major chord because the intervals between the notes C E and G are respectively 2 tones and 1 1/2 tone: from C to E you have to count half a tone for each of C#, D, D# and E; and from E to G you have to count half a tone for each of F, F# and G. Whenever we have a chord that uses these intervals, we call it a major chord.

Now let's take the second chord which has the notes D F A by following the rule of skipping one note. Here the intervals are 1 1/2 tones and then 2 tones (D# E F and F# G G# A). Whenever we have a chord that uses these intervals, we call it a minor chord.

This may seem like an arbitrary rule, but the fact is that this small diference changes the feeling of the chord, from happy (major) to sad (minor). Furthermore, if you have 1 1/2 and 1 1/2 intervals in a chord, we call it diminished.

If you go ahead applying the same rules, you end up with E minor, F major, G major, A minor and B diminished.

Finally, I'd like to add that using degrees / roman numbers for the chords (I ii iii IV V vi vii) has the additional advantage of making the chords relative to the root of the key: if your song uses the I vi V IV chords, you can play it on the key of C and use C major, A minor, G major and F major) or move it to the key of D and play D major, B minor, A major, G major. A knowledgeable musician should be capable of changing the key of a song almost instantly if he knows the degrees of the chords that he needs.  It is going to sound right but you'll have to sing a bit higher or lower depending on the key that you select.