Wikipedia has this to say about solar particle events:
A solar particle event (SPE), or "proton storm," occurs when particles (mostly protons) emitted by the Sun become accelerated to very high energies either close to the Sun during a solar flare or in interplanetary space by the shocks associated with coronal mass ejections.
Okay, so they seem to have to do with both solar flares and CMEs. And this page on part of the Goddard Space Flight Center website says:
The super-heated electrons from CMEs move along the magnetic field lines faster than the solar wind can flow. Rearrangement of the magnetic field, and solar flares may result in the formation of a shock that accelerates particles ahead of the CME loop.
So is that what a proton storm is, a piece of a coronal mass ejection that was accelerated beyond the rest of the CME by interaction with a solar flare? How does that work?
I used to think that SPEs were CMEs hitting Earth, but that doesn't seem to be the case, or at least only sometimes. After all, why have two terms if they are the same thing? I'm especially interested in it because of the highly variable time it takes for an SPE to reach the Earth. Under the right conditions, they move almost at the speed of light - there was one like that documented in 2005. So the way storms develop, and how that affects their speed, makes a big difference to how people and spacecraft need to be protected against them.