Background
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Let $\mathfrak{A}$, $\mathfrak{B}$, and $\mathfrak{C}$ be subcategories of the category of Banach spaces (over $\mathbb{R}$). Suppose we have a functor $\lambda:\mathfrak{A}^{op}\times\mathfrak{B}\to \mathfrak{C}$. 


Let $f:E'\to E$ be a morphism belonging to $\mathfrak{A}$, and let $g:F\to F'$ be a morphism belonging to $\mathfrak{B}$. (Note: These are morphisms of topological vector spaces). 


Then we have a map $$\matrix{Hom(E',E) \times Hom(F,F')\to Hom(\lambda(E,F),\lambda(E',F'))\\
(f,g)\mapsto\lambda(f,g)}$$


We say $\lambda$ is of class $C^p$ if for all manifolds $U$, and any two $C^p$ morphisms $U\to Hom(E',E)$ and $U\to Hom(F,F')$, the composition $$U\to Hom(E',E) \times Hom(F,F')\to Hom(\lambda(E,F),\lambda(E',F'))$$
is also of class $C^p$. (Note: We can replace $\mathfrak{A}$ and $\mathfrak{B}$ with categories of tuples to generalize this to several variables. In fact, this is what we do below.)


It is not hard to show that this induces a unique functor $$\lambda\_X:VB(X, \mathfrak{A})^{op}\times VB(X,\mathfrak{B})\to VB(X,\mathfrak{C}).$$ on vector bundles taking values in the appropriate vector bundle categories over $X$. 


We define a **tensor bundle of type $\mathbf{\lambda}$** on $X$ to be $\lambda\_X(TX)=\lambda\_X((TX,\dots,TX),(TX,\dots,TX))$, where $TX$ is the tangent bundle. 


However, this doesn't agree with the definition given on Wikipedia or anywhere else I've looked. 


Questions
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* Is this terminology nonstandard?
* Is the notion itself nonstandard?
* If the terminology is nonstandard, but the notion is standard, does it have a different name?
* Is this definition useful?
* Does this include more vector bundles as tensor bundles than the standard definition?



Is Lang's definition of a tensor bundle nonstandard?
Is the notion non-standard? As Emerton says, the answer is no, except perhaps in minor details.


Is the terminology non-standard (and more permissive than the usual notion of tensor)? Yes, I'd say it is. 


Because Lang allows arbitrary categories of Banach spaces, his notion is very general; by taking the categories small, one needn't have much functoriality at all. For instance, in the case where $\lambda$ has just one, covariant input, we can take $\mathfrak{B}$ to be the category with one object, $\mathbb{R}^n$, and morphisms $GL(n)$. Then $\lambda(\mathbb{R}^n)$ is a representation $V$ of $GL(n)$. On an $n$-manifold $X$, the resulting bundle $\lambda\_X(TX)$ is usually known as an associated bundle. Form the principal $GL(n)$-bundle $Fr\_X$ of frames (consisting of pairs of $x\in X$ with an isomorphism $T\_x X \to \mathbb{R}^n$). Then $\lambda\_X(TX)= Fr\_X \times\_{GL(n)}V$. 


The notion of associated bundle is one that differential geometers often find more convenient than Lang's, because it allows one to say exactly which structure groups of interest (here, $GL(n,\mathbb{R})$) and exactly which representations. 


A restrictive definition of tensor bundles allows those associated bundles where $V$ is a tensor product of copies of $\mathbb{R}^n$ and its dual. A broader (and I suspect fairly standard) definition allows subquotient representations of such - in particular, symmetric tensors and differential forms. But Lang apparently allows other things, e.g. $r$-densities (take the representation of $GL(n)\to \mathbb{R}^\times$ given by $g\mapsto |\det g|^r$).


Is the greater generality (compared to the standard notion of tensor) useful? In examples (e.g. densities) yes; in the abstract, not really - the interesting geometry is attached to specific classes of structure groups and their representations.


