A probably obvious observation could be that the shorter mode arises when the previous contact-arc events leave thin protrusions that can't support large amounts of current in the next one, which would make it interesting to check the behavior when a large amount of fine particles are present/put in the contact area, while still using distilled/deionized water. This might be easier to test by constraining the active volume with a test tube (e.g. immersed in the jar) rather than using the entire jar.
I wonder if the ringing being present only under dry conditions is somehow related with me hearing the resonant acoustic noise more easily when the electrodes are only partially immersed in the solution or just wetted with the acidic electrolyte. However, so far my AM radio seemed to pick up resonant RF noise only when the electrodes are partially immersed in water, even if a similar noise could be heard acoustically outside of it. (possibly this could be related with broadband noise overpowering the resonant noise on the AM radio under those conditions; it's not exactly a precision instrument).