Use an actual "submit" input element instead of a "button"

This has the advantage of allowing the user to just press the Enter
key on any field of the form in order to submit it. For this we move
the invocation of our javascript handler for the form submission from
onclick() of the button to onsubmit() of the form itself. Finally, we
have to reutnr false from our new onsubmit() handler so that the
browser doesn't reload the page (which would clear the message that
results from a new player being added).