Standard musical reference is given as: page/system (line/bar/stanza)/measure/beat (rarely used).
So this might be written as: 171/3/3 or more specifically 171/3/3/2.
When working an opera or musical where people know the music backwards and forwards, you can just shout out "171/3/3" and people will pick up where you're going from - yes, even when off-book. In shows I have worked on, chorus is expected to know where to be in this case, and dancers, principals, and orchestra need no additional pick up information. The SM must tell any tech departments (like tech or deck crew) where to reset and what to standby for. But mostly you can expect this level of musical awareness from a professional theatre.
Regarding the columns for the SM cheat-sheet, it sounds like a useful tool. But when you work with musicals a lot, you might not need this anymore. As you look at the pick up, you'll see blocking notes and can quickly scan your lights and other cues. This checking just becomes part of the SM running the tech rehearsal. This is always a moment of fluster, with everyone scrambling to find their place. It happens on the fly, but it's what we do. But when you know the show well, it's relatively simple to jump in right where everyone needs to be.
By the way, those numbers in the score that appear every now and then in boxes are rehearsal numbers. So your example would be referred to as "Rehearsal 56". This is different than page 56 or measure 56 - don't be confused! Rehearsal numbers are not used in musical placement notation - they're mostly used when rehearsing chorus and dancers, since the rehearsal numbers correspond to big sections of music. They are useful, but not terribly specific.