Riemannian metric with tensor products

Question

I have been studying differential geometry and I came across two different definitions for the Riemannian metric given by:

$$ds^2 = G_{ij} \, dx_i \, dx_j, \tag 1$$

$$ds^2 = G_{ij} \, dx^i \otimes dx^j. \tag 2 $$

Based on the above, I have a few questions:

1. What is the difference between them?

2. What are the advantages of the second definition?

3. What are the properties of the operation $\otimes$?

4. Does the fact of $dx^i$ and $dx^j$ being orthogonal imply $dx^i\otimes dx^j$ be zero?

5. Based on your vast experience, what are great introductory books that discuss the second definition in the framework of physics?

Thanks in advance.