Note name intervals are counted starting with the first note name. For example; the perfect 5th above C is G, and the count is C = 1 step, D = 2 steps, E = 3 steps, F = 4 steps, and G = 5 steps. The change of note name is 4 steps.
Semitone intervals are counted starting with zero which indicates no change. For example; the perfect 5th above C is G, and the count is C = 0 semitones, C# = 1 semitones, D = 2 semitones, D# = 3 semitones, E = 4 semitones, F = 5 semitones, F3 = 6 semitones, and G = 7 semitones.
The chart below lists many of the intervals as both Change of Note Name and Semitones.
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