For any two distinct integers $l,m \in \{1,\ldots, M\}$, let us define $\mathbb{P}_{lm}$; be the $(n-1)$-dimensional space consisting of vectors in $v \in \mathbb{R}_n$ satisfying $v^T(\mathbf{X}_l-\mathbf{X}_m) = 0$ Then as long as $\mathbf{X}_l-\mathbf{X}_m$ is nonzero the probability that a randomly chosen vector of norm 1 from ${\mathbb{R}}_n$ lands in $\mathbb{P}_{lm}$ is 0 (as it would be for any $(n-1)$-dimensional space).
So let $A$ be the randomly chosen matrix from $SO_3$. For any positive integers $l,m \le M$ and $p \le n$ only way $A\mathbf{X}_l$ and $A\mathbf{X}_m$ can agree on the $p$-th coordinate is if the $p$-th row $v_p$ of $A$ is in $\mathbb{P}_{lm}$. However, given $A$ randomly chosen from $SO_3$ according to the uniform distribution, then for any one such integer $p$, the $p$-th row $v_p$ of $A$ is also chosen according to the uniform distribution from the set of vectors of norm 1. The probability of $v_p$ so chosen being in $\mathbb{P}_{lm}$ is 0. Thus, for any one choice of $p,l,m$ the probability that the $p$-th coordinate of $A\mathbf{X}_l$ equals the $p$-th coordinate of $\mathbf{X}_m$ is 0. So finish using the Union Bound; only $nM^2$ such choices for $p,l,m$.