The stamps overprinted "Specimen" are not cinderellas at all. Just "normal" stamps aimed for a use other than postal. Early ones, were sent to the UPU headquartes in Bern for them to know the newly issued stamps. Lately, you can see "Specimen" stamps used for propaganda on a number of media. But, personally, I don't consider them "cincerellas".
From this time period, official overprints were for use with mail from governmental offices as would be expected. They are listed in the catalogs, you could buy mint sets. Most governments today would consider them unnecessary, and their issue has been considered a way for Nevis to get a few more bucks from collectors.
Aside from the UPU requirement, the specimen overprints were also used for publicity purposes and were not a gift made by a magazine. In that regard, they would be distributed by the post office for photos in ads and stamp magazines. Specimen sets would be given to larger direct purchasers of mint sets, as well as samples given to collectors, so they're not scarce at all.
In the 1970s I bought stamps directly from the postal service in St. Vincent. They would periodically mail me order forms for their new issues along with the stamps overprinted "specimen" for free. I always thought these specimen stamps were actually worth more than the original mint stamps since there were fewer of them than the real thing. The specimen stamps were not valid for postage and therefore are not listed in the Scott catalogue, so I guess their true value is unknown.