I ask “is there a website” and they responded with “yes, there’s an app”. To remove any assumptions from the conversation, why not respond with “Yes, there’s a website”?

This was for the management console of a router, right? In that case I'd be very hesitant to use the word "website". A "website" implies (at least for me) it's out on the internet, whereas an "app" doesn't necessarily imply that public access. I might have given a similar response to the guy you were on the phone with if I'd been in the same situation tbh.