dg.differential geometry - Is Lang's definition of a tensor bundle nonstandard?

This is a slight variation of the standard definition, as far as I can tell.

First of all, let me restrict to the finite dimensional context, since this is more standard.
Then a typical example of $lambda$ is the functor which sends $V_1,ldots,V_p,W_1,ldots,W_q$ to $V_1^{*}otimescdotsotimes V_p^{*}otimes W_1otimes cdots otimes W_q.$ The corresponding tensor bundle will be $(T_X^*)^{otimes p}otimes T_X^{otimes q}$ (here I mean the usual tensor product of bundles), and I can imagine sections of this being referred to as tensors of type $(p,q)$ classically. (It may be that the $p$ and $q$ would be reversed; I would check the conventions carefully of any reference that used terminology of this kind.)

You could check in Spivak, or on Wikipedia, or in any number of other sources to see the various kinds of terminology that are used for this construction, but whatever the terminology, sections of these sorts of bundles are precisely what are referred to as tensors in classical differential geometry.

In more modern treatments, you may see less of this terminology, because people will just write out explicitly the tensor products of bundles as I did above. But this terminology evolved over a long period of time, and tensors in differential geometry were being considered well before the functorial notion of tensor product of vector spaces was introduced.

As for how more general Lang's definition is, I can't think of any other $lambda$s of the top of my head (and it may be that, if you impose some natural axioms on $lambda$, there essentially are no further examples). As far as I can tell, he has simply abstracted the properties you need to have a functor of vector spaces give rise to a corresponding functor of vector bundles.

[Edit: As pointed out in the comment below, and in Tim Perutz's answer, there are indeed
other $lambda$'s: e.g., there are symmetric tensors and exterior tensors (the latter
giving differential forms, of course), which I mysteriously neglected when I wrote the above; one should certainly single them out, and my statement about there being no other $lambda$s is wrong as it stands --- all these
Schur-type functors are certainly candidate $lambda$s. One thing I hadn't realized, which is pointed out by Tim Perutz, is that the case of densities is also included in Lang's definition.]