Your equation fails to take into account what the original article (before I notified the author, who kindly responded and shouted out my blog without even asking for it) also failed to take into account.
When the dice rolls ":|" and you pass your turn, the state does change. Namely, whose turn it is changes. V(s, i, j) fails to capture this crucial detail.
This is accounted for because the second equation looks at sequences of two moves and not just one. After a sequence of two moves there are two possibilities. Either the state has changed or it has not. This is reflected in the left and right side of the second equation.
When the dice rolls ":|" and you pass your turn, the state does change. Namely, whose turn it is changes. V(s, i, j) fails to capture this crucial detail.