I think you have misunderstood the definition of "symbol." You should only take the term of highest order in the vector fields. Then the symbol is well defined.

More to the point, the symbol map isn't from differential operators to functions on the cotangent bundle, it's from the associated graded of differential operators for the order filtration to functions on the cotangent bundle. So, on operators of order less than n, you can do the operation you described to the highest order term, and you get something coordinate independent.

I think you have misunderstood the definition of "symbol." You should only take the term of highest order in the vector fields. Then the symbol is well defined. (EDIT: well, I guess I should have read Wikipedia first. I stick by my assertion that the symbol map one should consider is the leading order one).

More to the point, the symbol map isn't from differential operators to functions on the cotangent bundle, it's from the associated graded of differential operators for the order filtration to functions on the cotangent bundle. So, on operators of order less than n, you can do the operation you described to the highest order term, and you get something coordinate independent.