How to check which submission button was clicked?

The function submit_button() does generate:

<input type="submit" name="submit" id="submit" class="button button-primary" value="Save Changes"  />

As seen on the codex documentation page. Which gives you the possibility to check for a $_POST or, depending on the form, $_GET or generic $_REQUEST variable – to be exact for $_POST[ 'submit' ] or $_GET[ 'submit' ] or $_REQUEST[ 'submit' ]. The key for the variable is determined by the input tags name attribute value, which you can change for submit_button() by making use of the $name parameter.