I can't speak for what the majority of users here think of when they think of "clay." All I know is that natural clay is still alive and well, plenty of places still sell it and to me it seems weird to think that synthetic has become so ubiquitous that it is assumed
that one means synthetic when one says "clay"!
I have done a lot of wheel throwing and pottery, and as far as I know, natural is what is required there.
I like Laguna and Aardvark brand (natural) clays. Here's a promising-looking clay by Laguna: http://www.lagunaclay.com/clays/western/wc390.php