Difference between ionization and scattering of electrons

I am writing this answer to try and learn something about the various scattering processes for myself, so hopefully another more detailed and more sophisticated answer will also be posted, that both I (and the OP, of course!) can learn from.

Bhabha scattering is the electron-positron scattering interaction involving an electron and a positron :

${ e^{+}e^{-}rightarrow e^{+}e^{-}}$

The two-order Feynman diagrams describing this interaction are an annihilation process and a scattering process.

The Bhabha scattering rate is used as a luminosity monitor in electron-positron collider experiments. In scattering theory and accelerator physics, luminosity (L) is the ratio of the number of events detected (N) in a certain time (t) to the interaction cross-section (σ):

${ L={frac {1}{sigma }}{frac {dN}{dt}}}$

Luminosity values are useful in determine the performance and efficency of a particle accelerator, as the greater the integrated luminosity, the more data emerges from the (often expensive) experiment.

Ionization, (as I'm sure you already know, apologies)  is the mechanism by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons to form ions.  Ionization can result from the loss of an electron after collisions with subatomic particles, collisions with other atoms, molecules and ions, or through the interaction with light.

In contrast to Bhabha scattering , ${ e^{+}e^{-}rightarrow e^{+}e^{-}}$, Møller scattering ${ e^{-}e^{-}longrightarrow e^{-}e^{-}}$ denotes  electron-electron scattering. The electron interaction that is idealized in Møller scattering forms the theoretical basis of many familiar phenomena such as the repulsion of electrons in the helium atom.

Again, as I am sure you are aware, we need to include the both diagrams to complete the calculation of Møller scattering amplitudes. These illustrations give an indication of some of the terms included in the scattering amplitude, but to calculate a measurable quantity, we need to pick a reference frame, usually the COM frame, and assign helicities to the particles or average/sum over all possible spin states.