No, the probability distribution is the same. Watermarking changes the rng sequence used to pick from that distribution.
In reality, Gemini and Anthropic use SynthID watermarking which affects token probability distribution, i.e. their tournament sampling can pick lower-probability tokens which the LLM's distribution would otherwise not have. They likely use this over unbiased watermarking because SynthID is resistant against text edits.
Changing the distribution is the whole point: it introduces statistical regularities that can be detected.
Agreed.
> So the question is on average are these statistical regularities better or worse than those introduced by the alternative,
Indeed. This is an empirical question. That is what the article is about, for a specific setting.
> ... and the answer is no, if implemented properly.
I don't agree with that. This is not about implementation. It's about how the statistical regularities that are imposed on the full output distribution affect that distribution. There is a change - by construction. That change can be good in some situations and bad in others. The authors claim that it is mostly bad in the setting that they investigated. This looks like a fair statement.