You took a risk wiping down your violin with a damp cloth. Some varnishes do not like that, although I believe they are in a vanishingly small minority. I recommend for now that you just make sure you aren't getting your violin dirtier than it already is, and you can do this by: (1) using a soft microfiber cleaning cloth to wipe down your violin gently after each use to take care of loose rosin dust and the occasional fingerprint, and (2) handle your violin only by the neck and the button, so that you don't have grubby fingerprints all over the bouts and ribs that need wiped off.
Next time you have your violin at the luthier for some other work that involves removing the bridge, etc., you can get it professionally cleaned then.
Super Nikco is another product that you'll see. I believe this has a similar composition (perhaps an oil other than turpentine) but it includes an abrasive. Fiddlershop disclaims it as "intended for professional luthiers only" but it's readily for sale.
As Paul implies, cleaner basically contains varnish and solvent. In theory it removes rosin and replaces any varnish it removed with the rosin. Use it in microscopic amounts on tiny patches - fingertip on nearly dry cloth. Use it only where you identify rosin. Imagine you are restoring a Rubens painting, not currying a carthorse. Never use water on antique wood - you seem a bit too keen on that. Note that we don't know you. Is it possible you are a clean-freak?
Notice how good I am at theory when I have said I've never practised. Otoh, sometimes it's good to know enough theory to dissuade you from practice. I understand jumping off cliffs isn't healthy.
Microfiber cloth (sold by Shar, Johnson Strings, and others) is a good thing to keep around. Also use it to clean the stick of your bow.
One maker told me he uses mineral oil with a bit of toothpaste to clean varnish, but you're better off letting a pro get the serious gunk off. When it is really clean, the microfiber will hold you for quite a long time.
As is often pointed out, there are all kinds of cleaning products available, from household or workshop cleaning products to specially formulated varnish cleaners. Violin varnishes differ considerably, though, so a product that may work on one may destroy another.
Cream polishes tend to contain various abrasives that build up a sheen by cutting into the surface layer of varnish and scratching it until it gleams. Some are coarser than others, and they can take color off of a violin in short order.
Polishes that aren’t creamy usually contain solvents and oils. The solvents are intended to dissolve dirt, rosin, and grime and remove it from the surface, while the oils are intended as carriers so that the solvent doesn’t just strip the varnish off. However, sensitive varnishes can still be damaged by solvents even with a carrier. If the instrument has had any damage and the surface of the wood is exposed (e.g. open cracks, closed but unfilled cracks, nicks, scratches, or portions chipped or worn off), the cleaner will penetrate. Glue joints can fail and wood can be contaminated, which will compound the damage over time. Also, the oils used as carriers typically leave behind deposits that are near impossible to completely remove, and the deposits actually attract more dirt.
Detergents and degreasers attack dirt and oil and remove a lot of it, but they are not necessarily gentle on varnish, even if they aren’t abrading it. Some affect the pH balance enough to make the outer layers of the varnish turn gummy, which can lead to disaster quickly. Even the more neutral versions can be aggressive enough to wear into varnish.
There simply isn’t a cleaner or polish that works for everything, so any time you attempt to clean with one of them, you assume the risk of damaging your instrument further. This is why it’s so strongly recommended that the cleaning be done professionally, similar to the approach toward caring for paintings.
I feel like an “expert “ now.??
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However, if you can find it I like the Psarianos cleaner from Psarianos violins. I used to buy it from Concord music but they stopped carrying it. I eventually found it again since Psarianos violins sells it themselves, but I don't know if they still do. Last time I ordered it was a few years ago and I bought a few bottles of it just in case. I had to call them to place an order. It works very well and it's made by a luthier so I trust it.
Some people like Hill polish, but I wasn't a fan. I've heard good things about Laubach cleaner as well, but I haven't tried it myself. There's a YouTube video of someone showing them using it on their (not inexpensive) instruments and it looked like it did a good job. If her channel is still active I believe it's called the Violinista. I may buy a bottle and try it on my beater violin before I use it on my more expensive instruments, but I've been telling myself that for years now.