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> Have anybody ever noticed that cannon (in some sense) is also some kind
of piece, wich starts move as one, and continues as another?

Fairy-Max implements all its pieces this way: they always have a primary
and a secondary move, and on pieces that don't need that, these two are
simply taken the same. Primary moves kan be marked as 'hopping', in which
case the transition to the secondary phase takes place when it lands on an
obstacle with its primary move. Otherwise the transition to the secondary
phase is spontaneous, and can be configured to occur with a delay, i.e.
after making a specified number of steps of the primary move. This to
implement limited-range sliders, which first make a number of slider steps,
before reverting to a leaper for the final step.

In general all 'move rights' can change in the transition; not only the
'rider' right that allows repeating the step (if not blocked), but also
capture and non-capture rights. So a Cannon is defined as a hopper with a
Wazir step and rider rights, with primary non-capture and secondary capture
rights.

A difference with the interpretation given below is that in the Fairy-Max
description the transition takes place when the hopper is on the platform
square, not just behind it (Grasshopper) or before it (Contragrasshopper).
So the Grasshopper itself is in fact considered a transition piece, which
loses its rider rights in the transition, but gains capture + non-capture
rights, which it didn't have before.

Where exactly the transition takes place would only be noticeable if the
direction (actually board step) would differ between primary and secondary
step. A Mao is indeed implemented as a primary Wazir without rights and a
secondary Ferz with capture + non-capture rights, the transition taking
place spontaneously after the first Wazir step. Hoppers that alter
direction would be bifurcators.

The extension of Betza notation I proposed also uses 'chaining' of
different, independently specified phases, where the number of phases is
not limited to two (as it is in Fairy-Max). As there is also a notation
there for "indefinite repetition" of a phase, it would be possible to
define "multi-hoppers", like the mentioned Grasshopper-rider.