Isn't a tensor the generalization of scalar, vector matrix and so on? (PLUS the invariance under coordinate transforms?)

A box would be 3-dimensional indicating that tensors have 3 indices when in reality they have n-indices. Ir am i reading it wrong?

Which is really a roundabout way of saying a tensor is a multilinear relationship between arbitrary products of vectors and covectors. They're inherently geometric objects that don't depend on a choice of coordinate system. The box of numbers is just one way of looking at a tensor, like a matrix is to a linear transformation on a vector space

Where does this definition come from?

All the geometric definitions of tensors I have met always assumed a base, such that a change of coordinate or of parametrization would change the values of the tensor. Unless you define the tensor by its action instead of its values?