Viterbi algorithm recursive justification
The maximum is over the entire expression $\pi(k-1,w,u) \cdot q(v|w,u) \cdot e(x_k|v)$, for exactly the same reason that is worrying you. Moreover, under your original interpretation, there is no reason to keep $\pi(k-1,w,u)$ for the non-optimal $w$. We are forced to store $\pi(k-1,w,u)$ for all $w$ exactly for the reason you mention.