"People like you" versus "people like yourself"?

Solution 1:

You can find that yourself is

yourself, pronoun

¹ used when both the subject and object of the verb are you
Be careful with that knife or you'll cut yourself!

² used to give special attention to the subject of the sentence
Did you make the dress yourself?
You can do that yourself.

The second meaning is not reflective and does not require "you" to be previously mentioned. The meaning is simply emphasized, such as in:

not stressed

people like you

vs stressed

people like you yourself / people like yourself

EDIT:
Some further examples from Merriam-Webster's dictionary of English usage

A secret, kept from all the rest / Between yourself and me. - Lewis Carroll, "She's All My Fancy Painted Him," ca. 1854

Get me some good left-handers like yourself and Robinson - Robert Frost, letter, 23 Jan. 1921

In all this I look to nothing but the happiness of yourself, Mr. Randolph, and the dear children - Thomas Jefferson, letter, 27 Feb. 1809

Those who, like yourself, know what they are about - Walter W. Skeat, letter, in K. M. Elisabeth Murray, Caught in the Web of Words, 1977

EDIT2: I feel that choice of the first dictionary might have been unlucky, here is oxford, where I think the distinction is made perfectly clear:

¹ [reflexive] used to refer to the person being addressed as the object of a verb or preposition when they are also the subject of the clause

² [emphatic] you personally (used to emphasize the person being addressed)

The way I read the second meaning is completely unrelated to the first; under second meaning it is not reflexive, simply the meaning of yourself is you personally (try to substitute phrase "you personally" instead of "yourself" in the four examples from the MW; I think it does make it clear and very precise and I think that this is a very good definition of actual usage).

Solution 2:

"Yourself" could have been used in a different way. "Yourself" can be used as an emphatic appositive of you, as in "You yourself did it."
"Yourself" can also be used to refer to someone personally, like in the example you gave. However, "Yourself" is a reflexive pronoun, which means it must be preceded by a noun or pronoun which refers to it.

"You" is a pronoun that is used to address someone personally most of the time. Sometimes it is used to address several people, as in "You are all clever."

In this case, "youself" is being used in the same way as "you" : to refer to the person being talked to. There is no difference in meaning, and none of them are incorrect.