[cygx]

I'm aware. Though if we want to be more precise, that's about tensor fields, where the basis transformations of the underlying vector bundles (the tangent and cotangent bundle) are in turn induced by coordinate transformations of the base manifold.

However, physicists get introduced to tensors far earlier than any excursions into differential geometry when discussing rigid bodies.

[contravariant]

Yes I'd also call those tensor fields. The main point I'm trying to make is that the tensor transformation law only makes sense for such fields.

[cygx]

The terms co- and contravariant make sense on a purely algebraic basis, with components of tensors transforming 'the same as' or 'opposite to' the basis vectors. That the basis transformation is induced by transformations of some base manifold is incidental.

[ummonk]

Exactly. The fact that the bases are related to coordinates on the manifold is a property of differential geometry but the laws for transformation between bases are more general.