Do the hard plastic holders that the coins come in, similar to Air-Tites, have PVC? Should I change the holders to an Air-Tite as soon as I buy a coin?
No, hard plastic doesn't have PVC plasticizers that cause the damage. Some flexible plastic holders do not have PVC either as they are different type of plastic and are safe also. Solid PVC tubes as used in plumbing are safe also, many use them for containing valuable before burying them in their yard ( sealing them properly ). Air tites are good but expensive compared to generic , and really are not air tight.
I'm curious if you rubbed it with anything too. Before picture looked OK and the after looks very obviously scrubbed.
My local coin dealer said that cleaning cuts the value in half. I just noticed that this thread is from 2015 but for what its worth I didn't see an answer about the value so I'm adding what I've heard.
"Cleaning" is a complex subject. The general answer is to avoid cleaning coins. The example listed in this thread has residue from the plasticizer of PVC holders. It is acidic and should be removed using acetone. There is a lot more to it, but suffice to say, don't clean your coins.