What are the possible (and at least somewhat probable) particle interactions that could leave a signal in a cherenkov detector (such as super-k)?
One source suggested there would be
It also said that there would be "other interactions on $\rm^{16}O$ nuclei". If there are more than those two interactions, why is it those two that get talked about and not any of the others?
The thing is that any reaction which produces a charged particle traveling faster than local $c$ will produce Cherenkov radiation. In a neutrino experiment the two you list are the signal, and any others are the background. Someone else analyzing the same data from the same detector might not be interested in the neutrino signal and consider that background, but instead be interested in some other process.
I suppose the two reactions you list might generalize to oxygen nuclear targets: \begin{align} \rm \bar\nu_e + {}^{16}O &\to \rm {}^{16}N + e^+ \end{align} and \begin{align} \rm \nu_e + {}^{16}O &\to \rm \nu_e + {}^{16}O \\ &\to \rm \nu_e + {}^{15}O + n \\ &\to \rm \nu_e + {}^{15}N + p \end{align} The last two are sometimes called "quasi-elastic scattering": you can get pretty far by modeling it as a neutrino scattering from a free neutron or free proton, with fifteen "spectator" nucleons nearby.
By far the most common reaction in such a detector will be cosmic ray muons which pass through the water emitting Cherenkov light without scattering. The detector is built underneath a mountain to stop most of these muons, but some still leak through. It's common to find "veto detectors" outside of the Cherenkov volume to identify particles which originate outside of the detector.
You might also get reactions like muon-induced spallation, \begin{align} \rm \mu + {}^{16}O &\to\rm \mu + {}^{15}{O} + n \\ &\to\rm \mu + {}^{12}{C} + {}^4He \\ &\to\rm \mu + {}^{10}Be + {}^4He + 2p \end{align} I don't know the different cross sections for these, but I do know that production of beryllium-10 due to spallation in the air is commonly observed at electron accelerators. You can also have spallation on the heavier elements in the rock surrounding the detector produce neutrons, which leak through the veto region and capture in the water to produce a signal like the slow part of the inverse-beta-decay signal you describe above.
Similarly if there are radioactive elements in the water volume (in the metal of the walls, or the glass of the PMTs, or dissolved into the water) their decay products will generate Cherenkov light as well.
If you really want a comprehensive list, find a PhD dissertation from Super-K from your favorite university and look for a chapter with a title like "event selection and background suppression."
The only neutrino interactions known to be detected are those you provided: The charged current interaction and the neutral current interaction, respectively. These interactions can also be differentiated with respect to flavor.
In IceCube, for example, the charged-current interaction from a muon neutrino produces a "track" in the detector along the path of the muon (>100m). The charged-current of an electron neutrino produces a cascade similar to that of all-flavor neutral current interactions due to the short path of the products relative to the detector's size. I've heard people discussing neutrino-neutrino interactions, but certainly not in the sense that we could detect them, just that they would not be forbidden. A brief search for exotic neutrino interactions was unsuccessful.
There is no reason you can't have nuclear or nucleon excitation; or meson production reactions in the mix.
Reaction like $$ \nu + ^{16}\!\mathrm{O} \to \nu + ^{16}\!\mathrm{O}^* \,$$ with a subsequent decay of the excited nucleus or $$ \nu + n \to e^- + \Delta^+ \,,$$ or $$ \nu + p \to e^- + p + \pi^+ \,,$$ are accessible.
The three I have exhibited will be dependably detectable and can be removed from the data as noise or treated separately if there are enough. They should also be rare compared to the unornamented reaction.
Similar reactions are easy enough to find in the data set of LArTPCs where the high spacial resolution makes disambiguating the events easier than it is in a water Cerenkov machine.
