Most navigators avoid moon sights like the plague, too many corrections and too many vagaries.
However, if needs must, the following is the order of correction to be used for the most accuracy
Semi Dia (plus correction for augmentation)
Parallax _____________ (Horizontal parallax x Cos. App Alt)
Sorry, you will have to excuse the lack of alignment here.
The above is the most accurate in theory, in practice it is useless as you are talking couple of tenths of a minute against the quick and dirty method
In the above example, the Semi Diameter is taken from the daily pages of the almanac, and then corrected for augmentation (the effect that the moons diameter appears to change with altitude)
tables contain a table of the moons Total correction
In Norries Table, the arguments for total correction are apparent altitude against HP.
In the Nautical Almanac, it uses the same arguments, but a different layout to the tables.
The different methods all give slightly different results
However, in all the cases of total corrections that I have seen, the arguments are Apparent Alt against HP.
I would check the books
again and see if they were using different value's of HP.
With total corrections of 55.1 and 53.7, it looks like total correction method was used, but the HP was different