Background

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


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? 


I would say the answer to both questions is no. In fact, abelian varieties should be an "easy" case. For example, it is known that for abelian varieties (but not other varieties), the variational Hodge conjecture implies the Hodge conjecture. It is disconcerting that we can't prove the Hodge conjecture even for abelian varieties, even for abelian varieties of CM-type, and we can't even prove that the Hodge classes Weil described are algebraic. So if the Hodge conjecture was proved in one interesting case, e.g., abelian varieties, that would be a big boost.

Added: As follow up to Matt Emerton's answer, a proof that the Hodge conjecture for abelian varieties implies the Hodge conjecture for all varieties would (surely) also show that Deligne's theorem (that Hodge classes on abelian varieties are absolutely Hodge) implies the same statement for all varieties. But no such result is known (and would be extremely interesting).
