Is the axon part of a Schwann cell or no? Whenever I see diagrams of a neuron, it always shows the Shwann cells around the axon, but it doesn't seem as though anything distinguishes whether the axon is a component of the Schwann cells or not, or whether it's a component of the cell body itself (whatever that's called).

Your question isn't very clear, I think you have a bit of confusion in your use of terminology.

The axon is a part of a neuron, it's membrane is continuous with that of the cell body (also referred to as the 'soma'). Schwann cells are not neurons, they are a type of glial cell found in the peripheral nervous system. Schwann cells wrap their membrane around the outside of the axon to insulate it; this process or the presence of this insulation is called "myelination" and allows for faster transmission via saltatory conduction.

Cells that provide a similar function in the brain and spinal cord are called oligodendrocytes.