The rate of cooling of the glowing hot sphere occasionally appears to slow down or almost briefly stop. Sometimes it even briefly heats up again. I think this is called "recalescence" and is a known effect with cooling metals, although I wasn't aware that copper could be affected as well.
Earlier today I tried with a 0.35 mm high-carbon steel wire, and it appeared to behave similarly, but less clearly so. It looks like a condition is heating the material above its melting point. In the animation below, visual temperature (with camera settings constant) appears to suddenly drop as power (plasma) is removed, briefly stop, then decrease again.
It could still be related to hydrogen as its solubility in metals generally increases at their melting point; perhaps it gets absorbed and then released upon cooling. See graph here for example. However, as mentioned earlier, the above reaction is not strictly related with the plasma electrolysis reaction (below the water level).
In the previous comment I mentioned that for the cathodic plasma electrolysis, having a high water temperature helps significantly. Another major factor—it turns out, at least in my case—was having graphite in the solution. I'm not entirely sure of the mechanism, but without speculating too much it's possible it could be as simple as increasing the resistance of the electrolyte. Unexpectedly, also adding alcohol-based detergent helps, but it decomposes quickly due to heat from the plasma.
Keeping the electrode itself or the solution moving also helps to a limited extent, which should make sense if bubbles (from heat or else) around the electrode promote voltage breakdown.