The definition of symbol as presented in Wikipedia is not invariant — only the highest order terms. Some textbooks call those higher order terms symbols (Wikipedia suggests the name principal symbol), hence the Ben's answer, which refers to that definition.
The highest order terms are clearly most important for the properties of the differential equations, e.g. their positiveness allows to prove the existence of solutions (it's related to the fact that positively definite linear operators are invertible in linear algebra).
As for "Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism UL → SL.", this statement should be proved by induction order-by-order in a fixed coordinate system. It should be true in any coordinate system, but the homomorphism depends on it.
There is however, a canonical coordinate system given by the exp map, and this, I think (not sure here), is the canonical map referred to in the question.