I don't use actual wine corks.  I'm not much of a drinker anymore.  I think the last alcoholic thing I had was a dash of brandy in hot chocolate around xmas time 3 or 4 years ago.  LOL

I buy corks and sometimes sheets of the composite cork material from the craft store.  Useful for some musical instruments and assorted other projects.  Or hey, if you have any sort of a bottle that needs closed...

I'm not sure if this will clarify or add to the confusion.  But there's more than one kind of cork that can be called natural or "real".

One type is the plain cork material and has a sort of grain to it like wood.  It is a sort of bark, I guess that comes from a particular variety of oak that grows in some parts of the world. 

The other type is sort of shredded bits of cork (scrap, maybe?) that is pressed and probably glued together. 

I'd call the first solid, and the second composite, maybe. 

The one on the left is what I'm calling "composite" and the one on the right is what I'm calling "solid".

I have not found the composite to be very good for cleaning strings, since bits tear off from it a lot.  However, I was told that it is better for padding things (like where your chinrest touches the violin) since the solid has a grain like wood and can split under the pressure of something like a chinrest clamp.

So I usually keep some of both around. 

I started using cork for cleaning the strings back when I first started playing.  I had called an acquaintance who had played assorted bowed string instruments his whole life to ask for pointers and that was how he told me to clean the strings.   It works, so I never worried much about finding other ways.

Some thoughts on it, though.. It is a very old method.  While musicians certainly knew about distilled alcohol a couple hundred years ago, they used gut strings and there may be some important reasons why they used cork instead of whiskey or something to clean the strings.  Maybe alcohol would be bad for gut, dry it out too much or something.

I've noticed a lot of modern string players still use it though.  Might just be "tradition", or maybe it can do something undesirable to the synthetic cores that imitate gut, I really don't know for sure.

I still use it because I don't see where the cork can hurt anything and it does a decent enough job.  I could carry alcohol wipes in the little foil envelopes in my case and not have to worry about a bottle of alcohol leaking.. But they would get used up faster than a cork.  I'd have to use a new wipe every time I clean my strings, which is every day.  Cork does eventually get too cut up to work well, but it takes quite a while.  The small one you see in the pic is one I've been using for about a month now.

I'm also a klutz, and sooner or later would end up dripping some alcohol where it shouldn't go, and that would suck.  "You just need to be real careful" may as well be a recipe for eventual disaster for folks like me.  LOL 

I can't think of a real good reason to take more chances than I have to with the instrument.

Another thought, corks were a very common throw-away item in the past.  The bottle cap of their day.  So they would have been very easy to get, and it may have been thrifty to use a throw-away item for cleaning or might have been less bother than having to wash rosin out of a hanky or something.  Or maybe there is some reason I don't know, that makes it a much better choice to use cork on gut strings and the usage has just carried over to other types of strings as they were developed. 

Anyway, if you look at the cork in the right side of the pic, you can see a notch in it, near the bottom.  I just cut a small V shaped notch a couple mm deep, so I can get the underside of the strings easily when cleaning with a cork.  That notch has deepened from the wear as I've used it.  Out of a small cork like that, I can usually get a couple months of daily cleaning before what is left of the cork is too small to be useful. 

And as has been mentioned, if you use the cork method, you want to hold the violin sideways so any rosin dust or cork crumbs that you're taking off don't fall on the instrument or into the Florentine holes.

I don't know as it matters hugely what method one uses to clean the strings.  So long as you don't let rosin cake up and harden all over them, because that can definitely do bad things to the sound.