How to know what submit button the user clicked?

Question

The submit_button() function is a wrapper for get_submit_button(). Now that function has multiple arguments, but the most interesting for your actual problem is the 3rd argument name. It sets the HTML name attribute.

<form action="">
    <input ... etc.
    <button type="submit" name="choice-a">
</form>

Now everytime you process your form with an empty action attribute, you will point to the current request http://example.com/wp-admin/example.php. There you will be able to fetch your arguments via the super globals $_POST or $_REQUEST (which simplified is mostly a combination of $_GET and $_POST).

Then just inspect the array and do whatever you need/want to do.