The definition of symbol as presented in Wikipedia is not invariant — only the highest order terms. Some textbooks call those higher order terms symbols, hence the Ben's answer.
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 given by the exp map, I think (not sure here).