If I change the response on the Submit page for one form, does that change only show on that form and not the others

Thats about it. 

I believe that you are referring to the Thank you message on your JotForm. If so, the answer is 'Yes' and you are right. Each JotForm has its own Thank you page and it is required to change the message for each form that you want to modify the message.

Hope this information helps!