One moment that always confused me about Twilight.

How exactly did Edward expect to die at the end of new moon?
I believe his plan was that he would expose himself to the sun in that one town when they were having a parade or something(its been awhile since Ive read the books)and then the vampires would kill him for exposing himself to the world therefore the world would know about vampires.
But my problem is that would imply that the people of the world would think a shiny person is a vampire. But if what I read is true then the way people perceive vampires is incorrect to Twilights vampires.
Because the people of that world think that vampires die when sunlight touches them.