I don't want to speak for Brian on this, but the impression I got from the questions posted above seemed to indicate a more general idea of "magic" than the more rigorous definitions indicated in Dale's and Bick's posts.
Strictly speaking, of course, what the elves, etc., do seldom comes under the rubric of "magic"; but (as Galadriel herself indicates when speaking to Sam before using her mirror) to hobbits (and, by extension, to the readers who are, after all, given a view of the world "as seen by the hobbits") such things would qualify because they are rooted in a spiritual or supernatural realm. Thus, both sorcery (including necromancy, one of the major types of "magic" used by the Enemy), which is a complete violation of the natural order (hence carrying much of that idea of "sin" which Machen made use of) and the more benign types of summonings, healings, callings on the Powers, and so forth which are a "natural" inheritance of the Elves, tend to fall into this category. Speaking more strictly: these are the fantastic elements of the story; the things which separate it from, say, science fiction, which -- at least purportedly -- bases its happenings on an extension or extrapolation of a scientific (or at least pseudoscientific) idea or view of the world.
In Tolkien, it seems to me, there is a tension between the natural world of the everyday, and the incursion of the spiritual (whether sacred or corrupt) which carefully balances his work between the poles of a naturalistic view and that of an universe where a godhead is still very much present, albeit at times seemingly remote. Of course, the involvement of such a godhead also implies a similar involvement of its opposite; and it is the tension between these elements (naturalism/supernaturalism; a godhead and its enemy, etc.) which provides much not only of the narrative drive, but the power of his vision by deliberately posing without resolving the uneasy balance between the "mundane" and the "magical". As the hobbits themselves become increasingly aware of the presence of this element in a world they view as (essentially) "prosy", so do we; with the result that, following the aesthetic views he set forth in "On Fairy-Stories", the reader is given a regenerated view of the world in all its magic possibilities, whether they be blatant or subtle.
As I say, I could be mistaken on this, but this is what I gather Brian was asking about when he speaks of the "magic" of LotR and the related volumes.