The american R sound

Take a look at the information I link to on this page, which I have titled The geography of America's dialects and cultural affinities compared.

In particular, visit the links for the dialect maps produced by Rick Aschmann, where you will find both audio and video clips illustrating how various sounds are typically pronounced in different regions of the United States. The page depicting his principal dialect map contains numerous supplementary sections that discuss particular linguistic features of American English. Among them is a section on R-coloring which is directly relevant to your question.

I can safely say that both Aschmann's summary of R-coloring and the clips I mentioned will demonstrate that the woman you referred to is thoroughly mistaken in asserting the existence of a so-called 'correct' way to pronounce the sound of the letter R in American English. There are in fact many different ways that Americans sound that letter; it would not be reasonable to demand that they should change their pronunciation merely to suit her preference.

I would say both are correct, but the “wrong” one she describes is less common in most accents than the “right” one. I usually hear the “wrong” version overused by speakers of languages with an R-like sound in which the tip of the tongue comes in contact, or nearly in contact, with the top teeth—for example, Hindi or Japanese.

The only situation in which I think I might use the “wrong” version is in a phrase like “you’re running”, a situation where this woman would compress her lips.