Not True - the Interval is the rest in between (half-time), as stated in 90% of Soccer/football matches. Terminology is culturally specific and sport specific - Americans don't "own" terms, and Interval described by other sources (Gerschler/Reindl or Igloi)) is just as valid.
As HRE stated - many, especially Americans, would recognize Intervals as being the Reps themselves (see "Once a runner"), even if that is not the case, but what else would you expect from a country who still uses the imperial system of measurement. Of course (as with rowing) one could call the repeats "pieces". Jack Daniels confuses it even more by distinguishing between Intervals and Repetitions which are both "repeats" of some particular intensity - the former with controlled recovery and the latter with more recovery or at least less controlled. Therefore I prefer to use the word recovery rather than interval, for the sake of simplicity, as it is less confusing.
As to recoveries - many programs are vague about it, and often don't distinguish between the various types - standing, walking, jogging, floats and based on distance, time, HR or lactate - which changes the dynamics of the session. Tegen had a complex mode of jog/rest/jog, which makes some physiological sense as well. Of course a complex training session incorporating a number of physiological parameters could incorporate all types of recovery as well.