I think when some say a trained voice sounds "natural" they mean it doesn't have that classically-trained, operatic style.   That probably has more to do with the pronunciation of various vowels and with the position of the mouth, etc., than with things like relaxation of the throat, breath support, projection, etc.

Don't most professional singers these days work with vocal coaches, whether they're singing pop, rock, jazz, folk, or opera? At least those who perform a lot and plan to keep their vocal instrument healthy over the long haul?

Other than that, I'm curious as to just what distinguishes "the natural voice" from other voices.