When black has a winning strategy, black already applies "zugzwang" for white's very first move: Black only wins because white has to make a move. If white could skip, black would not win.
> Edit: it may be that the statement "without Zugzwang" implicitly (or perhaps by definition) means you are allowed to skip moves?
Yes. It's not well defined, but I'd say a non-zugzwang win is a win (or rather a winning position) where you would also win when your opponent can skip their turn. A zugzwang win is a win that is not a non-zugzwang win.
So chess being a win for one side is equivalent to starting position being zugzwang for the other side.
It's obvious now, but so interesting to me, I never thought about it that way! Thanks for taking time for explaining yourself.
> Then white would win (and in particular not lose) by skipping the very first move and then playing blacks strategy
I understand the other comment, that there do exist setups in which colors can be effectively switched by e.g. 1. e3 e5 2. e4, but that requires cooperation on black's part. How does white "skip" the first move? Thanks in advance.
Edit: it may be that the statement "without Zugzwang" implicitly (or perhaps by definition) means you are allowed to skip moves? If so, that clarifies my confusion.