The headline is wrong. They don't claim Gemini was trained on docs.
" It also seemed to be able to regurgitate accurate information related to unreleased mechanics that the developer says they had only written into GDocs the day before the player’s AI interaction."
If it was written only the day before there's no way the model was trained on that data. The claim is Gemini accessed his docs while generating the answer. We know Gemini can access public docs so I think the question here is if it accesses private docs in a breach of security, the doc had wrong permissions or it was all just a coincidence.