Merriam-Webster has as much linguistic authority in the Commonwealth as the OED has in the US, which is to say none. And to be honest the OED is increasingly modish and unreliable even for British use, though obviously still superior, because we've got to argue about something, right?

I stand by my original verdict. A hamburger is a sandwich in a similar way to how a Victoria sponge is a sandwich. It is indeed technically a sandwich, but for any practical purpose in real life, its membership of the sandwich family is irrelevant.