Grey Gardens offers an interesting example of this. When Ms. Resnik plays Little Edie in the second act she employs two modes of singing. When Edie is singing diegetically (to Jerry in "The House We Live In", or to the Maysles Brothers/Audience in "The Revolutionary Costume for Today"), she sings employing her thick New England accent. She brilliantly sacrifices the 'beauty' in the song for the sake of her character. But, when she sings extradiegetically in "Another Winter in a Summer Town" she sings it beautifully (in fact, really beautifully), and we can presume her voice is now unencumbered by her neuroses or her region and she sings from the soul.
This is an amazingly effective choice and the kind of choice that shows the potential and the power (if, still, the sentimentality) of musical theater, and the rewards granted by taking the additional leap its genre demands of the audience. To make beauty and ugliness an aesthetic choice is an uncommon position in this more conventional world, but it pays off for the performance and for the form.
It is convenient, perhaps, that we are to understand her soul's voice as beautiful. Taking this line of argument to its extreme it would be interesting to hear what Polpot's voice (for instance) would sound like in an extradiegetic context (or how this would be applied to Sondheim's Assassins). The real case may be that Ms. Resnik is simply singing 'neutrally' (luckily, this means beautifully) when her character is exempt from narrative demands. Either way, I do think it is important in Musicals to differentiate these separate modes of singing as a means to examine their significance to the character, the narrative, and (of course) the audience.