Representing chained XOR operations as linear inequalities
Introducing another integer variable $t$, you can express the condition $x_1 \oplus x_2 \oplus \cdots \oplus x_n = 1$ as in the following canonical form.
$$x_1 + x_2 + \dots + x_n - 2t \le 1$$
$$- x_1 - x_2 - \dots - x_n + 2t \le -1$$
$$(x_1, x_2, \cdots, x_n) \le (1,1,\cdots,1)$$
$$(x_1, x_2, \cdots, x_n, t) \ge \mathbf 0$$
$$(x_1, x_2, \cdots, x_n, t) \in \Bbb Z^{n+1}$$
You can also replace the first two inequalities by the following equality.
 $$x_1 + x_2 + \dots + x_n - 2t = 1$$
In other words, you can actually eliminate one of $x_1, x_2, \cdots, x_n$ by replacing it with $t$. 

Note $t$ is an integer not bounded from above. However, you can also bounded it by the following inequality, which is implied by the existing inequalities.

$$0\le t\le \lceil \frac{n}2\rceil-1$$



The idea is that $x_1 \oplus x_2 \oplus \cdots \oplus x_n = 1$ if and only if the number of $x_i$'s that are 1 is odd. This answer is essentially the same as the answer you mentioned.