Chatbprs are genAI. Any artificial intelligence like NPCs, autopilot, playing games against the machine, playing chess against the machine... All of those have been called AI.

GenAI is a subset where what the AI does is generate text or images instead of taking a deterministic option. GenAI describes pretty well what it does generate a text or image output, no matter the accuracy of the text. The AI is optimised to generate output that looks like what you would expect with the given input, and generally it does exactly that, even if it hallucinated facts to fit the idea of the response that they are supposed to give with the given input.