What makes two solutions distinct in the Vinogradov mean value?

Question

As in e.g. https://en.wikipedia.org/wiki/Vinogradov%27s_mean-value_theorem, the Vinogradov mean value $J_{s,k}(X)$ is defined to be the 'number of solutions' to the system of $k$ simultaneous Diophantine equations in $2s$ variables given by $$x_1^j+x_2^j+\ldots+x_s^j=y_1^j+y_2^j+\ldots+y_s^j$$ for $1\leq j \leq k$, with $$1\leq x_i,y_i \leq X.$$ What I don't understand is exactly what counts as distinct solutions. Clearly, given a solution one can swaps the $x_i$ with the $y_i$ or permute the $x_i$ or $y_i$ to get 'new' solutions. But the trivial lower bound for this quantity is supposedly $\lfloor X\rfloor^s$, 'taking $x_i=y_i$', suggesting that these solutions don't count, but doesn't this still count permutations of the $x_i$? I'm unsure exact what the 'number of solutions' means here.