When using the forms webpart to embed a form in a page, there is a redirect URL. I don't really want to send the user anywhere, and having the redirect url set as the current URL of the page makes the page reload and doesn't really give the user confirmation of their submission. I'd like to have the form just show a little "thanks for your submission" type message within the frame on the page. I did add a confirmation message to the "Save" button which pops up at the top of the page, but then the page still reloads.
I see that there is a "Confirmation Page" setting for On-Prem, but don't see anything comparable for 365. Am I missing something?
As far as I know - not. You can try with the other type of a button - "Save and Continue" - this will not redirect user to any location, but after submission will keep him on the form. However the form will reload in the meantime.