Why is there no deterministic algorithm for determining the majority vote of a recursive 3 branch tree without inspecting all the leafs?
Yes, you are misunderstanding the algorithm. While I explain your misunderstanding below, in general, when such a misunderstanding occurs, see if you understand what the algorithm is trying to do, and then try to match your understanding with what is written.

While there are numerous errors in textbooks, usually they are slight and easy to correct. Whenever an error seems to have occurred (which doesn't seem to be the case here), see if you can try to correct it by making some small change to the argument.

In this case, the first ancestor of the second leaf which has at most one sibling determined is the leaf itself, which has one sibling determined. According to the strategy, our answer to the second leaf would be complementary to our answer to the first leaf.

Your misunderstanding might stem from the fact that for you a node is not considered its own ancestor. However, for this strategy to work, a node must be considered its own ancestor, as your example shows. Presumably the text defined ancestor correctly to include the node itself, but even if it doesn't you could correct the proof by changing slightly the meaning of ancestor. This demonstrates my points above regarding reading mathematical texts.