general relativity - What is the relationship between the formal definition of a tensor and the frequently discussed notion of a "higher order matrix"?

I've been doing some self study on the principles of tensors & manifolds in preparation for a first course in general relativity. I tend to learn better when presented with the full mathematical formalism of a topic, and to that end I've been pursuing more mathematically rigorous texts (Geometrical Methods of Mathematical Physics by Schutz, Tensor Analysis on Manifolds by Bishop, and the first few chapters of Wald's General Relativity).

All of these books define a tensor as follows (with some slight variation):

Definition 1 Let $V$ be a vector space over the field $\mathbb{F},$ and let $V^*$ be its dual space. A tensor on $V$ is a multilinear map $T : V^* \times V^* \times \dots \times V^* \times V \times V \times ... \times V \rightarrow \mathbb{F}$.

This makes perfect sense, and the following math is understandable using this definition.

The question is this: I frequently see people explain tensors as "like higher order vectors." I have seen more than once the following claim:

"Definition" 2 A scalar is a 0th-order tensor, a vector is a 1st-order tensor, a matrix is a 2nd-order tensor, and you can keep going from there, thinking of tensors as an extension of the concept of a matrix.

So the question is how are these two definitions related?

Using the first definition, a 1st order tensor would be, presumably, a linear map $T : V \rightarrow \mathbb{F}$. And I suppose such a linear map $T : V \rightarrow \mathbb{F}$ is a vector in the dual space of $V$, so that does match up with the second definition's claim that a first order tensor is a vector. But it doesn't sit quite right with what the second definition seems to be implying.

This second definition raises all sorts of other questions: certain sets of matrices can be made into vector spaces. So what distinguishes a matrix and a vector in the wording of the second definition?

Is this second definition perhaps a special case, where vectors and matrices of real numbers correspond to tensors over specific vector fields? Or is it something else entirely?