I would say it is better to think of x as the basic process of awareness - the brain~mind activity that animals have too. So really the story is y~x.
{I'd say that x is the physically measurable state and is objectively measurable.}
Then the extra x* issue is self-awareness. The ability to introspect "objectively" on conscious states.
{yes, x* includes self-awareness, but it also includes all the mental phenomena that we associate with consciousness that some call qualia, some call experience or phenomenal experience, and some call feeling}
Introspection is of course a learned socialised habit, not an innate "hardware" feature.
{Partly true, but introspection could equally arise without socialization as it would seem to follow from consciousness, so I wouldn't say that introspection is strictly something learned. I suspect someone growing up on an island with no one around would discover introspection independently and very quickly}
http://en.wikipedia.org/wiki/Lev_Vygotsky
http://en.wikipedia.org/wiki/George_Herbert_Mead
{not sure why you referenced these. when providing references, it would be nice to provide your thoughts about a topic, and if those thoughts would benefit from further reading, then providing references would be wonderful, and well accepted.}
And also x* would not be epiphenomenal. Except in a certain sense.
The socialisation of the human brain through the self-regulatory mechanism of language is in fact a good example of downward causation - constraint exerted from a cultural level to the individual level.
{I think this is an excellent example of how we differ in our views and why I've said earlier that we are in 'different camps'. I don't see this example as being one of causation, except in a very loose sense. It's similar to the statement in economics that says, "The cost of goods and services increases with increasing demand." This confuses physical causes (such as the conservation of energy, mass and momentum laws) with very generalized causes that have meaning to us as humans. Human meaning provides us with a wonderful, meaningful and comprehensive understanding of the world around us, but we must resist the temptation of thinking of these concepts as physical causes. They are not physical causes. This is actually nicely explored in Bedau's paper where he talks about "gliders" and the game of life. These things we see as gliders are weakly emergent, but it helps us as humans to understand their behavior. There is no downward causation in the behavior of a glider.}
Society teaches you to mind your manners, pursue certain goals, think in particular ways. The causality is from the global scale to the local so that you in your own head are negotating your needs vs the social needs.
See http://www.dichotomistic.com/mind_readings_JCS%20freewill%20article.html
That is of course why Sautoy was reacting with such feigned horror to the notion he had no freewill and his brain was deciding up to 10 seconds ahead of time. Society demands we be in control of our bodies. That is society's need - even if it is a fiction and leads to naive statements about the nature of consciousness.
{let's leave free will and the Leibniz delay out of this thread.}