Why are 'differential operators on manifolds' differential operators?

Consider the following operators:

1. Gradient - Acts on functions and return vector fields. On $M = \mathbb{R}^3$, it has the signature $\nabla \colon \Gamma(M,M \times \mathbb{R}) \rightarrow \Gamma(M,TM)$.
2. Directional derivative of some vector field with respect to a fixed vector field $X$. Acts on vector fields and return vector fields. On $M = \mathbb{R}^3$, it has the signature $\nabla_X \colon \Gamma(M,TM) \rightarrow \Gamma(M,TM)$.
3. Divergence - Acts on vector field and return functions. On $M = \mathbb{R}^3$, it has the signature $\mathrm{div} \colon \Gamma(M,TM) \rightarrow \Gamma(M,M \times \mathbb{R})$.

They are all classical first order differential operators with different domains and codomains that can be interpreted as acting on and returning sections of vector bundles. On $M = \mathbb{R}^3$, all the vector bundles involved (and in general) are trivial and so you can just pick some global isomorphism $TM \cong M \times \mathbb{R}^3$ and think of the operators as operators with signature $D \colon C^{\infty}(\mathbb{R}^3,\mathbb{R}^n) \rightarrow C^{\infty}(\mathbb{R}^3,\mathbb{R}^m)$ for appropriate $n$ and $m$ (they act on vector valued functions and return vector valued functions). However, working with arbitrary manifolds, the notion of a vector field which was before (identified with) just a tuple of smooth function (an element of $C^{\infty}(\mathbb{R}^3,\mathbb{R}^3)$) now generalizes to a section of a possibly non-trivial vector bundle $TM$ and so the operators now should be generalized to operators that act on sections of (possibly non-trivial) vector bundles. And indeed,

1. Gradient is replaced/generalized with the differential of a function $d \colon \Gamma(M, M \times \mathbb{R}) \rightarrow \Gamma(M, T^{*}M)$ (function $\mapsto$ covector field) or, in the presence of a Riemannian metric, by a gradient operator $\nabla \colon \Gamma(M,M\times \mathbb{R}) \rightarrow \Gamma(M,TM)$ (function $\mapsto$ vector field).
2. The directional derivative is generalized by the notion of a covariant derivative (also called an affine connection) denoted by $\nabla_X \colon \Gamma(M,TM) \rightarrow \Gamma(M,TM)$.
3. Divergence is generalized in the presence of a Riemannian metric to an operator $\mathrm{div} \colon \Gamma(M,TM) \rightarrow \Gamma(M,M\times \mathbb{R})$.

First of all, note that your definition of a differential operator includes as a special case the case when both $E$ and $F$ are both the trivial rank one bundle $M \times \mathbb{R}$, in which case sections of $E$ and $F$ are just real-valued functions on $M$.

Also note that on $\mathbb{R}^n$, one might be interested in differential operators that act on vector-valued functions. Vector-valued functions on $\mathbb{R}^n$ are really, in fancy terminology, sections of a trivial vector bundle over $\mathbb{R}^n$. So it is natural to generalize this to sections of more general (possibly nontrivial) vector bundles on manifolds.

In my (limited) experience, one reason that differential operators between vector bundles (e.g., the exterior bundle of differential forms) are important is because some of them (e.g., Laplace- and Dirac-type operators) are closely connected to the geometry and topology of $M$.