The claims of NYT are more than just about training. They're claiming that as part of the ChatGPT software; it _looks up_ stuff in a database of articles. That is beyond _training_.
If a human kept around a briefcase of NYT articles they didn't pay for and let you view them for a fee I think everybody would agree that's copyright infringement.
The focus on "whats the difference between an AI and a Human learning" is a great slight of hand to bypass the obvious copyright infringement OpenAI is doing.