I think it might be as well be the memory stored in the brain itself. The brain is huge. There is not only the conscious in it but lots and lots of unconscious things too. Saying "the" brain "knows" the limb is no longer there is a bit like saying your computer knows the right English orthography. The information is somewhere in there but that doesn't mean it's automatically used in all cases.

When I type what I'm used to typing instead of what I intend to type I also love to say my "fingers" remember it but really I think it is stored in some part of the brain.

Perhaps, the heart or blood is the hard drive which contains the information and the brain is the processor of this info.  

Origin of phantom limb pain needs to be addressed at two levels - Central & Perpheral.

Central : The fact that Phantom Limb pain has been reported from children born without limbs indicates the role of brain in the genesis of symptoms.It is known that a specified area of the cerebral cortex is delegated  for reception as well as transmission of nerve impulses to and from various parts of the body. These neural pathways get integrated during the development of the fetus. Loss or arrested development of a body part leads to the lack of sensory input from the affected part leading to  altered nervous activity in the Brain which is perceived and projected to have originated in that limb.

Peripheral :Damage to the the sensory nerves in the affected limb causes disruption and subsequent disorganized regeneration of nerve endings in the stump, leading to abnormal discharges by the sensory neurons which again will be perceived by the brain to have originated in the area originally represented by these nerves.

Therefore, in case of Phantom Limbs, it is altered neural activity - both in the Brain as well as at peripheral level, which is at play rather than the memory.

Musicians and dancers push beyond the limits of the conscious - they practice until they can't get it wrong. That means they're running on automatic while planning the shopping. There are two stages, training the muscles to do things they don't normally do, and training the subconscious to process the triggers - sheet music, other players, the music - to perform the piece. It's a similar exercise typing, we don't think how to type, we just do it. Similarly with talking, or any number of other activities. The proof that it's the brain is that pianists suffering from strokes lose the ability to play: if it was just the body they'd still be able either to play given the "Go" trigger, or not if they can't handle the trigger, but they display impaired ability to perform, an ability but not the same as used to be.