We analyse alternative extensions of stable models for non-disjunctive logic programs with arbitrary Boolean formulas in the body, and examine two semantic properties. The first property, we call atom definability, allows one to replace any expression in rule bodies by an auxiliary atom defined by a single rule. The second property, well-supportedness, was introduced by Fages and dictates that it must be possible to establish a derivation ordering for all true atoms in a stable model so that self-supportedness is not allowed. We start from a generic fixpoint definition for well-supportedness that deals with: (1) a monotonic basis, for which we consider the whole range of intermediate logics; and (2), an assumption function, that determines which type of negated formulas can be added as defaults. Assuming that we take the strongest underlying logic in such a case, we show that only Equilibrium Logic satisfies both atom definability and strict well-suportedness.