[cygx]

There are no coordinates to transform in an ordinary tensor space and therefore no way for a tensor to be affected by such a transformation.

Sure there are: Any basis of the underlying vector space(s) induces a basis of the tensor space. Components respective to some basis are coordinates. You can then investigate what happens to the induced basis (or rather, the respective components) under a basis transformation of the underlying vector space(s), which is where the "physicist's" definition of tensors originates.

[contravariant]

The components of a vector aren't the same as the coordinates physicists talk about when they're dealing with tensors. The components would be something like the value of the magnetic potential, or the local wind speed. The coordinates would be the location where that particular vector is 'anchored'.

A change of coordinates does indeed induce a change of basis, but a change of basis isn't really a change of coordinates. And strictly speaking some vector spaces don't really have an obvious basis (without invoking choice), so having a basis be a prerequisite for the definition is not ideal.

The whole requirement that a tensor is 'something that transforms like [...] under a coordinate transformation' is just how physicists have chosen to phrase that a vector bundle is only well defined if it's definition isn't dependent on some arbitrary choice of coordinates. In my opinion this requirement is more easily apparent in the mathematical definition where there is no choice of coordinates in the first place, rather than the physicists way of working with some choice of coordinates and checking how things transform.

[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.