Were you able to solve the problem, Alex? I have the same problem as you. My button is named correctly ("submit") but the form keeps adding new rows to the database.
Some use the following code:
if($submit)
[enter the data]
But this doesn't work for me. I think the problem is that the form is on the same page and, as you said, redirects the data to itself. Of course, I could use a separate file for the form, but I just refuse to believe that there is no solution to this problem.