All the other intervals have a major and a minor version. The major scale only uses some of these intervals and the minor scale uses others.
I think this might be a bit misleading. Every major scale has every interval from m2 to M7, but it depends on where you start the interval.
The quoted link gives a lot of different information, so it is hard to know what in particular you find to be misleading.
But I'll hazard a guess since you referred to major scales in particular.
of intervals has been derived from major scales, always from the tonic up. Thus:
C to same C = P unison
CD = M2
CE = M3
CF = P4
CG = P5
CA = M6
CB = M7
C to next highest C = P8
They are all majors unless they are perfects. These also give us the measure of how many semitones they are apart - you could consider semitones to be sort of like "inches", like a smallest unit of measure.
Any interval that is a semitone smaller than these majors will be minors; two semitones smaller = diminished; one semitone higher = augmented. With the perfects you can have a diminished or augmented. The reason is historical.
Yes, you can have a minor interval within a major key. DF involves two notes that exist in the key of C major, and DF is m3 (minor 3). But if you use the system of considering the bottom note to be the tonic in order to name the interval, in D major the higher note is F#, DF# = M3, a semitone lower DF = m3, so the system still holds true. It's probably easier to know how many semitones = m3 (three) or M4 (four) and go by that. Or sound, if you know what a major 3rd or minor 3rd sounds like.