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tl;dr: Having one logical choice for a notation that is consistent with the rest of one's definitions is not the same as *defining* that notation to mean that choice. Until one has done so, the notation has no meaning (in a strict reading of a math text).

In this excerpt, Lang is defining, presumably for convenience, two notations for the same concept of the product of a finite collection of elements from a monoid. One of those notations is $\prod_{\nu=1}^n  x_\nu$, and the other notation is $x_1 \cdots x_n$. (Neither of these definitions explicitly invokes the concept of a general indexing set, just the natural numbers.)

But both notations are defined via recursion on the structure of $x_1 \cdots x_n$, and the base case of that recursion is $x_1 \cdot x_2$ (i.e., the monoid operation itself). So the empty product (the reduction of $x_1 \cdots x_n$ to the 0-element case) is undefined at this point, as is the meaning of $\prod_{\nu=1}^0 x_\nu$. If Lang plans to use either of those cases in the rest of the book, he had better say what they mean, so he does. As you note, he is constrained by the associativity rule he just laid out, so if those notations are to have any conventional meaning, they had better mean what he's defined them to mean; but he technically also has the option of simply excluding those cases from having any meaning at all, which is why one would say they're convention rather than being strict consequences of his existing definitions.

As for whether one can be derived from the other, again, we're establishing notations here, and in the initial definition of these notations Lang stipulates that $n > 1$. So the equivalence between the two notations for $n \le 1$ is not yet established.

(In fact, going off of just this excerpt (my copy of Lang is hiding in a box somewhere right now), Lang seems to leave the $n = 1$ case in an undefined limbo! You can say that *if* $\prod_{\nu=1}^1 x_\nu$ is to have a meaning consistent with the above definitions, it had better mean $x_1$, but you can't say that it *does* mean that with a strict reading of these definitions.)

I should probably also just affirm that in *informal* conversation, this distinction is rarely drawn so finely. If you were to introduce some new notation by sketching the equivalent of just the first line of math in this excerpt on a napkin or blackboard, I would expect nearly all mathematicians to fill in the blanks without you having to be explicit about it. I just wouldn't recommend it for your thesis.