Starting slowly with distilled water is great for dirt, etc. Unfortunately, acetone or distilled water does not remove corrosion. I'm still curious as to what you would do next to the OPs 2c after you used both.
It's almost always recommended not to clean your coins. At the first sight of satisfaction that it looks better, STOP. My opinion only, good luck.
That's a good coin to practice cleaning on. Technically the coin isn't worth much. The 1865 is one of the most common 2 cent coins. In it's current condition it would end up in a dealers 25 cent box. Play with and post a picture showing what it looks like after cleaning.
I would soak separately in an old pill bottle for a period (could be weeks) where I would shake it occasionally to free/lift loosened debris. I would change out the distilled water at some point(s) to keep it clear and to note any progress. I've not tried the olive oil trick, but if I were to, I'd do it after I felt that distilled water was not working any longer. Normally at that point I would put some Verdicare on it to sit for a bit...usually only an hour or less on copper...perhaps longer on clad/silver coinage. While the Verdicare is puddled on the coin, I would use a soft bristled toothbrush (cleaned between coins/uses) to gently lift any debris likely to continue attacking the substrate (e.g., stubborn verdigris). Folks might be surprised by the results of distilled water soaks and some agitation in an old pill bottle. I kept reading how the Ancient enthusiasts were soaking for extended periods with excellent results. It works better than tap water, perhaps because it's free of impurities that counteract some of the effects. Once I noted how cloudy the pill bottle water had gotten (pre shaking), I knew something was different. So I use it regularly now on metal detecting finds I'm curious about.
All restoration methods involve usually a q tip. The coin presented is a problem. Because it might actually need something stronger like a rose thorn. There is nothing wrong with starting out light and working towards a stronger emulsifier. Olive oil from my experience needs a stronger approach. Like rubbing the coin between your fingers. It worked to get almost all the scrolling visible on this Spanish American war medal. This one was a mess when I dug it up. Sorry I don't have any before photos.