So far, the actual distance, in half-steps, between the two notes has not mattered. But a third made up of three half-steps sounds different from a third made up of four half-steps. And a fifth made up of seven half-steps sounds very different from one of only six half-steps. So in the second step of identifying an interval, clef , key signature, and accidentals become important.
Listen to the differences in the thirds and the fifths in the above Figure 4.30.
So the second step to naming an interval is to classify it based on the number of Half Steps and Whole Steps in the interval. Familiarity with the chromatic scale is necessary to do this accurately.