This usage is indeed very common, but it is always meant and understood hyperbolically, and the meaning of the word relied on by the hyperbole is just M-W's sense 1.

To see this, remove the word "literally" from such a sentence. You are generally left with a statement that is still hyperbole or metaphor and not literally true. Which word are you going to redefine in a sentence like "It turned the world upside down." to enable you to interpret it literally? If you understand it as hyperbole, why does adding the word "literally" require any special meaning for that word?

Conversely, if you believe that the word "literally" literally can mean "not literally", in the same way that, say, "cleave" is an auto-antonym, it would authorize statements like

  (*) When you use metaphor, you are speaking literally.
  (*) What I said was the plain truth, but John took it literally.

which I think you will agree are not standard usage.

Moreover, if you replace the word "literally" in a sentence like

  "It literally turned the world upside down."

with something that means "not literally", like

  "It figuratively turned the world upside down."

it changes the meaning. The first sentence is hyperbole and the second is awkwardly literal.

I understand the desire of lexicographers to document this usage, which is as you say common (and legitimate) and seems to confuse people, but it's just not a different meaning of the word.