In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?

Let $\mathcal{E} \subset J^{2k}E$ be the submanifold (provided that this subset is a submanifold) of all $2k$-jets sitting in the zero-level set of $\rho(E(\mathscr{L}))$, the PDE submanifold. This is Definition 8 in my [arXiv:1211.1914]:

A PDE submanifold $\mathcal{E} \subset J^{l}E$ is called linear if $E\to M$ is a vector bundle and $\mathcal{E} \to M$ is a vector sub-bundle of the $J^{l}E \to M$ vector bundle. The PDE system is called quasilinear if $\mathcal{E} \to J^{l-1}E$ is an affine sub-bundle of the affine bundle $J^{l}E \to J^{l-1}E$.

In my notation all the bundle projections (as well as their restrictions to sub-bundles) like $E\to M$ and the naturally induced $J^{l}E\to M$ or $J^{l}E\to J^{l-1}E$ are anonymous. One also needs to recall that the projection $J^{l}E \to J^{l-1}E$ is naturally an affine bundle.

That definition is immediately followed by the obvious Lemma 1:

A quasilinear PDE submanifold $\mathcal{E} \subset J^{l}E$ can always be represented as the kernel of a morphism of affine bundles. Namely, there exists a vector bundle $F\to M$ (naturally interpreted also as an affine bundle) and a morphism $f$ such that the following diagram of affine bundle morphisms is fiber-wise exact: $$\require{AMScd} \begin{CD} \mathcal{E} @>>> J^{l}E @>\xrightarrow{~0~}>f> F \\ @VVV @VVV @VVV \\ J^{l-1}E @= J^{l-1}E @>>> M \end{CD}$$

Now, I don't think that I had found the same exact definition anywhere else (that's why I don't give any reference), but I think that it's the one that makes the most sense.

Igor gave a coordinate independent definition of quasilinear equations which the Euler-Lagrange equations satisfy. Still missing is a definition of quasilinear differential operator, which the Euler-Lagrange operator satisfies.

You already observed that

$M\times_M V^\circledast E$ is generally not a vector or affine bundle over $M$.

hence the definition of quasilinear operator found in Bryant, Chern et. al. on p. 397 (which agrees with the definition found in Krasil'shchik, Lychagin, Vinogradov, Geometry of jet spaces and nonlinear partial differential equations p.160) cannot be applied directly, since they assume vector bundles over $M$. But $V^\circledast E$ is a vector bundle over $E$, and the fix might be to expand the definition of nonlinear DO:

Definition: Let $V\to E$ be a vector bundle over $E$ and $\psi:J^k E \to V$ a morphism of fiber bundles over $E$ (not neccessarily linear, since we don't assume a linear structure on $E\to M$), then the map $\psi \circ j^k$ is called a nonlinear DO. Call such an operator quasilinear if $\psi$ is affine, when restricted to any fiber of $J^k E \to J^{k-1} E$.

I haven't checked, but suspect that the E-L-operators satisfies this definition of quasilinear DO (with $V=V^\circledast E$). Observe that the zero set of a quasilinear $\psi$ is a quasilinear PDE $\mathcal{E}\subset J^k E$, as in Igors answer. Observe also that a nonlinear DO in the more restrictive sense of: a map of fiber bundles $\rho: J^k E \to W$ over $M$ is a special case of the previous definition by pulling back $W$ to $E$.

Note also that such a generalization of DO is unavoidable in cases like minimal surfaces, where we don't have a fiber bundle structure on the ambient space $E$. (In fact, in the case of jets of submanifolds we probably need to allow vector bundles $V$ over $J^1 E$ instead of over $E$, since the vertical bundle $VE$ doesn't exist and is replaced by a "normal" bundle.)