This is the difference between using individual corrections and Total Correction.
The method I outlined first is using the individual corrections to reduce the moon sight, and it may not be explained in either book you are using.
But, a bit of background.
I use the Hydrograhic Office Nautical Almanac.
This Almanac is published on an annually, and contains the daily/hourly values for GHA, SHA, Declination etc for Sun, Moon, planets, and Aries
(dont worry about Aries).
For the moon, it also contains the hourly value of Horizontal Parallax.
Moons value of parallax is maximum when it is on the horizon. It is a correction that adjusts for the angle measured at the observer to the angle at the center of the earth.
As the moons altitude increases, the value of parallax decreases. When the moon is at 90 degree (overhead) parallax is zero, the earths centre, the oberver and the moon are all on the same line.
The definition I had to learn was
HP : Angle at the bodies centre subtended by the observer and the centre of the earth when the body is on the sensible horizon
To find parallax at any value of altitude, the maths is
Parallax = HP x Cosine Apparent Altitude
Another individual correction is for the moons semi diameter.'
As the moon is close to the earth, there is the effect that to the observer, the moon is further away when on the horizon, and closer when overhead. To allow for this, there is a correction called Augmentation of the Moons Semi Diameter.
This correction may be listed in various Nautical Tables.
But, to make moon sights a bit easier, you have the TOTAL CORRECTION which takes all of the above into account. It might make a difference of a couple tenths of a mile, but in the real world, no big deal.
If you want to exercise the grey matter, then you can download The American Practical Navigator (Bowditch). It will all be in there, and far more than you need, but it is good back ground stuff
Can find it here
Maritime Safety Information
and good luck. If you have trouble sleeping, this book will help