About the notation of XOR-SAT
XOR is an associative operation, so $x_i \oplus x_j \oplus x_k$ is unambiguous. There are in fact four satisfying assignments, given by
$$ (x_i,x_j,x_k) \in \{(0,0,1),(0,1,0),(1,0,0),(1,1,1)\}. $$

XOR satisfies the useful property
$$ (-1)^{x \oplus y} = (-1)^x (-1)^y. $$
Therefore $x_i \oplus x_j \oplus x_k = 1$ is the same as
$$ (-1)^{x_i} (-1)^{x_j} (-1)^{x_k} = (-1)^1.$$
If we think of our variables as $X_i = (-1)^{x_i}$, then the equation $x_i \oplus x_j \oplus x_k = 1$ is completely equivalent to $X_i X_j X_k = -1$. Since XOR is really just addition in the group $\mathbb{Z}_2$, you can think of both formulations as linear equations.



Yet another interpretation of XOR is in terms of sets. Just as AND corresponds to intersection and OR to union, the XOR operation corresponds to symmetric difference, sometimes denoted $\triangle$ or $\Delta$. You can easily check that $x_S x_T = x_{S \triangle T}$, essentially using $x_i^2 = 1$.