Why are there so many versions of the "Black Madonna of Częstochowa"

Well, to begin with, the images linked aren't really 'versions' of the original. They'd be more accurately called as copies. They all feature the same fleur-de-lis robes, brown complexion - even the scars on the cheek are similar.

As for the third 'version' - it simply looks like a copy of the original (or perhaps the original itself), covered with a 'riza'. A riza is a metal cover meant to protect and adorn an icon, often made with precious metals. It's a common feature for Eastern Orthodox Icons and is also found in some icons venerated by Roman Catholics.

An example of another riza-covered icon venerated in the Roman Catholic tradition is Our Lady of the Gate of Dawn in Lithuania. Here is the image without its riza, and here is a photo of it covered with its riza. So it's possible that the third photo you linked is the original (or a copy thereof) simply covered with a riza.

Without properly knowing the source of these images linked, it's near impossible to make a definitive pronouncement as to which is the original, especially since the image in question is a famous image that has had many copies of it painted.

It really isn't uncommon for famous icons and images to be copied, especially reputedly miraculous icons such as Our Lady of Czestochowa. But I do think that it's incorrect to say that there are many versions - rather, there are many copies (which may feature slight deviations, depending on the painter's skill), but they nevertheless show and incorporate the same features as the original.