You should do it like this/that. (Like is a preposition. This is a pronoun. Like this is a prepositional phrase that modifies do. This points to an something at hand; that points to something far
I think the person using like this could be mimicking the procedure or pointing to a cookbook, and the person using this way could also be doing either. The two expressions are very similar, as these definitions show:
likeprep.1. Possessing the characteristics of; resembling closely; similar to.
I'm confused, too. I don't see any difference at all between "this way" and "like this" in the contexts you have presented, i.e., adverbial. Of course there's a difference when they are adjectival:
"I want another book like this." (Correct) "I want another book this way." (Incorrect)
Up until now, we've been discussing "Like this/Like that" in the context of "in this manner". Generally speaking if you can replace "like this,that,etc" with "in this manner", you can use any phrase. Such is not the case with "I want another book like this." where the "like this" is a short form of "like this one". Hope this clarifies, somewhat...