Of course it's ill-defined; every genre is ill-defined. This S&S discussion is nothing compared to some of the flamewars I've seen over whether fantasy is a subset of SF, or whether SF stands for "science" or "speculative".
But I'll keep trying. To me it's not exactly access to magic that defines S&S but prevalence. Magic is everywhere in Middle-Earth--rings, swords, phials, cloaks, doors, maps, horns, elves, and forests are all overtly magical there, and Gandalf calls it forth visibly with a word or gesture or pinecone. In Earthsea wizards and sorcerers are commonplace enough that you could go find one if you needed to.
Conversely Elric is one of just a handful of sorcerers that you run into in his entire career, and even then his 'magic' is limited to Stormbringer and summoning powerful extradimensional beings, the latter only with difficulty or at great cost. In Hyboria, sorcerers are very rare, usually up to no good, and typically found in forgotten ruins. In Lankhmar magic is virtually nonexistent except for the prophecy and coincidence of Ningauble and Sheelba.