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This is a really well done explanation of the various type of pieces that have been invented so far. While missing out on a few advanced concepts (mostly from large Shogi variants), it covers all the common (and much of the uncommon) variables very well.

Under movement, a stepper and multiple steppers deserve some mention. Single step pieces, like the King, Commoner, various Shogi Generals, Fers, Wazir, etc. are really a class all thier own. While they can be interpreted as either short range riders or leapers, they aren't really either.

Multi steppers, such as the Chu Shogi Lion make several moves as single steps. Other multi steppers are a sort of limited range rider. They can only stop at thier far square, but must pass through the other squares on the way. The Xiang Qi Horse is a bent piece of this sort.

Some mention of the special subcase of rider, the slider, should be made. A slider moves by multible steps (like a Rook, Bishop, Queen, or Lance). A non-sliding rider moves by multiple leaps (like the Knightrider and other oblique riders).

Under capture, you should mention that some peices (again from obscure Shogi variants) can capture friendly peices as well.

Some mention of restricted movement zones could be useful as well. The most well known examples are from Xiang Qi, where the Elephant cannot cross the river and the General and Advisors cannot leave the palace.

A related concept from Janggi (or Changgi), Korean chess, is enhanced movement zones. Various pieces have additional powers of movement when in the palace.

I'm sure there are some other things that are missing as well, but that's what I've noticed at this time.

Movement: define a 2-stepper PIECE to make 1 or 2 moves like that PIECE. The direction of movement can change with every move, unlike the case of a rider.

I use the name 'Commoner' for the usual (Wazir+Ferz) compound. The Lion has all the moves of a 2-stepper Commoner, with no restrictions. It can make a null move by returning to the square it started on. It can capture by igui. It can even make two captures in one turn. The Lion also can make a single Squirrel leap. In Jetan the Chief is a 3-stepper Commoner, which always makes exactly 3 steps, plus some other restrictions.

My recent games use the name 'War Elephant' for the 2-stepper Ferz, which has also been called Jester and Free Padwar. None of these chess variants allow the piece to make a null move. Again, the original definition in Jetan was most likely a Chained Padwar, moving exactly two steps.

I have a vague memory (several years old) of a 2-stepper Knight in a some chess variant. It may have been allowed to capture by igui. Not to be confused with the Ubi-Ubi. EDIT: The SUPER KNIGHT '... can capture a maximum of two enemy pieces in one move.' in Lim Ther Peng's Supremo Superchess.

The Bent Hero and Bent Shaman in Joe Joyce's Lemurian Shatranj are 'Inclusive Compound' pieces. They can be regarded as a '2-steppers' where the only 2-step moves allowed are those of two different pieces.

This essay is an admirable start! I think we can expand on the categories quite a bit and provide more examples. As the author himself says:

'I would encourage an effort to build on this, so that a more complete resource would be available to Chess Variant designers or problemists.'

I think there may be an obvious category which is 'Imitator' as in the Chameleon / Mimotaur. [Edited addition: I see now that David has written an entirely separate and also excellent essay on imitators.]

The 'Ooze' has a cousin in the Amoeba from Hedden's Microorganism Chess which I think introduces some more original categories of pieces as well...