There are as many different ways of applying a finish with a pad as
there are products to use. I have, for years, polished this way.
To my mind, "French polishing" refers to the final steps to the finish,
whereas padding refers to the intermediate steps to the final finish.
A French polish is actually a final finish in itself, but I suppose
there were finishers who would rottenstone and oil that finish also.
There are as many methods as there are finishers, and everybody has
their own "proprietary method" and formula. Also, it will help nobody
to itemize one's own ingredients because of the differences in methods,
climates, humidity, and the timing and sequences of the procedures,
based on these variable factors.
Quickly, padding "smears" the lacquer or varnish into the pores of the
wood and builds the finish with as little material as possible into a
filled base in order to be final-polished. It is the padded finish
that will have to be "sanded" or flattened, because filling pores still
leaves streaks that can be seen through the French polishing to follow
later. But that also depends on the wood. For instance, certain
mahogany, like a "blister-figure" and "flame-grain" mahogany won't
leave streaks, whereas Honduras and Philippine will.
The constituency of the pad also varies with the procedure, and one
depends on the other. Padding is an art, and must be taught. It
cannot be read about and then perfect results ensue. Even the addition
of the oil and what is used for oil is important, because the oil
becomes a part of the finish. If the oil will not harden with time,
then you don't want to use it.