I'm a beginner at Atiyah-Singer index theorem and I've reviewed some results about theorem. Here's some questions.
Ive seen the topological index is equal to $$\operatorname{ch}(D) \operatorname{Td}(X)[X]=\int_{X} \operatorname{ch}(D) \operatorname{Td}(X)$$for any elliptic differential operators.
But I see that the topological index can also be defined for Dirac operator $D$ of a Clifford module $\mathscr{E}$ over a compact, oriented manifold $X$ of even dimension which is $$\int_{X} \widehat{A}(M) \operatorname{ch}(\mathscr{E} / S)=\langle[\widehat{A}(X) \operatorname{ch}(\mathscr{E} / S)],[X]\rangle$$
My question: It seems that the former one is defined on any elliptic differential operators and this is defined by $K$-theory? The latter one is defined for Dirac operators (which is also elliptic differential operators), for spin manifold, and the method is different by using heat kernel proof.
$(1)$ Can someone explains me why these two definitions coherent and in which situation?
$(2)$ I can't see why there is statement that the latter one(by using heat kernel proof) is much general than the former one(by using original $K$-theory proof)?