The answer is once-over. Milady describes it directly: the once-over shave requires less time for a complete shave service and should result in a smooth face without being a close shave.
The book adds historical context — the once-over shave was popular when men patronized barbershops daily for their shave, since it doesn't require the extra time and irritation risk of a true close shave. The second-time-over and close shave are separate, more thorough shave types covered elsewhere in the same section.
Milady Standard Barbering, 6th ed., Ch. 13 (Shaving and Facial-Hair Design), p. 349: "The once-over shave requires less time for a complete shave service... This shave should result in a smooth face without being a close shave."
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