A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: David C
Date: 2020 May 14, 01:35 -0700
It would appear that while the USPS method reduces the sight by the Law of Cosines, while the Hilaire method uses Haversines.
Are there other difference
You are confusing two different things - the method used to plot a line of position (LOP) and the method used to reduce the sight.
Marq St Hilaire devised the intercept method in which an intercept and an azimuth are used to locate a LOP.
There are various ways of reducing the sight. The Law of Cosines is one and the haversine formula another. Note that the haversine formula was devised decades after St Hilaire did his work. The sights can also be reduced with what are called short tables (such as HO28 Dreisonstok) and lookup tables (such as 24 & 229).