The naming system for intervals assigns each an ordinal number (second, third, fourth, fifth, sixth, seventh) relative to the first or starting note. The numbering stems from the fact of having seven alphabetical letters in use for musical notes. A – B – C – D – E – F - G. Any interval describing the distance between A and C must be some type of third since making A the first letter automatically fixes C as the third letter.
Each interval is also assigned a quality.
The unison, fourth, fifth and octave are perfect and the second, third, sixth and seventh are major. Any interval whose distance from the start note is one semitone more than these perfect and major intervals is described as augmented.
Any interval whose distance is one semitone less than a major interval is called minor. For major and minor intervals, a simplistic view would be to think of them as ‘big’ versions / ‘small’ versions of the same lettered note.
Intervals one semitone smaller than both perfect and minor intervals are called diminished.
The major third interval is four semitones.
The perfect fourth is five semitones.
If you reduce the length of a perfect fourth it will become four semitones and its name will be based on having shortened the perfect fourth - hence it will be called diminished fourth.
Correct.
The perfect fourth (you state fourth only without including the perfect but I assume you mean that) is five semitones.
The major third is four semitones.
If you increase the length of a major third by one semitone it will become five semitones and its name will be based on having extended the major third - hence augmented third.
Correct (but you don’t use major in the name any more).
See this topic for some more: Intervals, scale degrees and more