Here are my conclusions on this matter. First, diagonal is a geometric term
that has nothing at all to do with specific kinds of changes in
coordinates. On a 2D board, the meaning of diagonal is unambiguous. It
describes movement that runs through opposite corners of a space. In 3D
and higher dimensional games, it becomes ambiguous, because there are
different kinds of diagonal movement. In a 3D game, you can distinguish
between diagonal movement that runs through the vertices of 3D spaces,
what Parton calls vertexel, as well as diagonal movement that runs through
opposite corners formed by two edges instead of three. It is appropriate
to distinguish these two kinds of diagonal movement as bi-diagonal and
tri-diagonal. Contrary to what I said earlier, tri-diagonal does not mean
triaxial, and it does not mean triaxial and diagonal. Rather, it describes
the geometric property of movement that runs through opposite vertices of
a polyhedron. It is a useful term for any multidimensional game beyond 2D.
For 4D and up, we can add tetra-diagonal, penta-diagonal, etc.

As distinguished from tri-diagonal movement, bi-diagonal movement runs
through opposite corners formed by only two sides. The Bishops in Chess,
Raumschach, and Hexagonal Chess are all bi-diagonal movers. Thus, Bishop
is an appropriate name for the piece which has it in both Raumschach and
Hexagonal Chess. Gilman has described a property shared by the Bishops in
Chess and Raumschach but not Hexagonal Chess. What he has described is the
property of uniform biaxial movement, not the property of bi-diagonal
movement.

My main point has been that there are two different methods of describing
piece movement, and each method should have its own terminology.
Orthogonal, bi-diagonal, and tri-diagonal are all geometric terms.
Confusion has resulted, because the mathematical method has not had its
own terminology, and people who have used it have tried to redefine the
geometrical terms in terms of coordinate math instead of in terms of
geometry. Case in point is Gilman, who was using diagonal to mean
uniformly biaxial. The mathematical method is a perfectly valid way of
describing piece movement. It just needs its own terminology, which is why
I have proposed the terms uniaxial, biaxial, and triaxial.

As for the word triagonal, I have no problem with the concept behind it,
but I do think that compressing tri-diagonal to triagonal obscures its
meaning. Instead of contrasting triagonal with diagonal, which is like
contrasting British with European, we should contrast tri-diagonal with
bi-diagonal. Tri-diagonal is not a kind of 3D movement that merely
resembles true diagonal movement. Rather, it is a specific kind of
diagonal movement and should be more clearly acknowledged as such.

As for orthogonal movement, it can be understood as straight movement that
never passes through corners. All types of diagonal movement pass through
corners, and all types of angular movement pass through corners, but
orthogonal movement never passes through corners.