There are 3 types of failure I've seen.
Primary failure, is what we use in HST. Slowing reps, or losing form, although losing form would usually come later in a set.
Secondary failure, is when you simply can't do another rep, used in most failure training programs.
Total failure is when you can't move the bar more than 2 or 3 inches. Used in MuscleNow and Mentzer type training. Preferably referred to as 'psychotic neural overload stress point'.