Yes, that definition is fine for machine learning, but it's not quite complete for physics. To extend your definition for physics, a tensor is a function that takes an ordered set of N row vectors and M column vectors as arguments and spits back a real, coordinate-invariant number as a result.
I think you get coordinate invariance for free if you think of a vector as an object in its own right, rather than as a tuple in a coordinate system. But then I guess it's more accurate to speak of vectors and covectors than row vectors and column vectors.
