denotional semantic for while - fixed points
Your proposed alternate definition is no good.  It tries to define $[[\text{while b do S}]]$ in terms of $[[\text{while b do S}]]$.  That's a circular definition: you can't define something in terms of itself.

"FIX" typically refers to the least fixpoint, and is a way of avoiding circularity.  See your textbook for background on fixpoints and the precise definition of the "FIX" operator.