RECONING: is the process of repairing a speaker that has a burnt out voice coil.  it's called reconing as everything including the cone needs to be replaced, because of the way the cone is permanently attached to the voice coil, spider, & surround.
RE-EDGING:  the process of replacing the surround (the suspension between cone & frame).
REFOAMING: is the process of repairing a speaker's foam surround. Foam surrounds are  subject to more damage and deterioration than other surrounds,  and therefore get a special form/distinct name/description of repair of their own.  The foam is replaced, as well as the dust cap (in better quality repairs), which allows for shims to be used.
SHIMS: shims are thin pieces of material (usually plastic) used to space the voice coil and pole piece during speaker assembly.
SURROUND:  the supporting structure between the frame and the cone.  It's what holds the cone's edge centered and to the frame.  It can be made of various materials: foam, cloth, paper, rubber.