Tensor: for some people, a tensor is an element of a tensor product of vector spaces. For others it’s a section of a tensor product of tangent and cotangent bundles on a manifold. Members of the first group would call that latter notion a tensor field.

More interestingly:

Tensor rank: a tensor of rank $r$ is either a sum of $r$ simple tensor (outer products) or an element of a tensor product of $r$ vector spaces (an “$r$-dimensional array of numbers”). For example, a matrix is a rank $2$ tensor in the latter sense.

I speculate that at least 25% of Math Stack Exchange questions about tensors are about confusion over one or both of the above.

Tensor: for some people, a tensor is an element of a tensor product of vector spaces. For others it’s a section of a tensor product of tangent and cotangent bundles on a manifold. Members of the first group would call that latter notion a tensor field.

More interestingly:

Tensor rank: a tensor of rank $r$ is either a sum of $r$ simple tensors (outer products) or an element of a tensor product of $r$ vector spaces (an “$r$-dimensional array of numbers”). For example, a matrix is a rank $2$ tensor in the latter sense.

I speculate that at least 25% of Math Stack Exchange questions about tensors are about confusion over one or both of the above.