Anthropic could put a big flashing warning text at the top of every chat that says “We are spying on you and will report anything scary to the police!” and it would not make any difference in this case.

You can call it anything you like, but only the legal definitions matter for the legal case.

After working on several court cases about surveillance activities, I'm definitely aware that whether I call something spying or not has little relationship to whether courts will think it's legal.