I'd like to add a few thoughts on this topic. If you search the www you can find a range of US paperweight artists have produced "Artist proof" paperweights and signed them either "Artist Proof" or just AP. For example a Correia Art Glass Tropical Fish paperweight marked ARTIST PROOF (from 1989); Jim Donofrio paperweight of wizard faces and flowers signed Jim Donofrio 02 artist proof; Rick Ayotte 2001 artist proof hummingbird and orchid miniature paperweight; Cathie Richardson Green Octopus paperweight identified by the seller as an artist's proof; and a 1976 Paul J. Stankard Artist's Proof Mixed Bouquet Paperweight - an experimental bouquet paperweight signed by Paul J. Stankard and marked "AP1 23876." In fact there is a book called "Paul Joseph Stankard: The First Decade" which is a catalogue of artist proof paperweights by Stankard in the collection of the Wheaton Museum of Glass.
What is an "artist proof" paperweight. It is the first of a design produced by an artist and intended for use as a model for a series or as a test for a design to be used again in other weights. By its nature, the original artists proof is unique even though the design is then reproduced as a series. I don't think there is any reason to think it is less than perfect or damaged or not finished off completely. Some may be, but there is no reason to expect this. The US "artist proof" paperweights mentioned above sold for thousands of dollars. Some artists may choose to leave their "artist proof" paperweights unfinished but I do not think that is the norm.
A few months ago there were several "artist proof" paperweights from Caithness Glass which were offered for sale on ebay. Unfortunately I did not manage to buy one. They each had documentation that they were original artists proofs and/or experimental designs. Caithness Glass have been taken over a number of times in recent years, and ever since they became part of the same organisation as Edinburgh Crystal, all kinds of items have been put up for sale that were never sold to the public before. I have bought items which the sales staff told me "We weren't allowed to sell these before, but now it seems that everything is being cleared out of the stock room". So it did not surprise me to see artists proof paperweights and experimental designs coming out of Caithness Glass.
The late Colin Terris (Caithness chief designer for many many years) described the making of the first model for a series of paperweights. Once the design was accepted, that artists proof was then the model for the whole series, and any paperweights which were too large or too small or where the design was not close enough to the original model, were rejected and if they were nevertheless good paperweights, they were etched CIIG and sold as seconds only from the Caithness shop (not via their official retail outlets). These paperweights often turn up on ebay, as Frank has pointed out. The point of this story is that Caithness in particular must have had literally hundreds of artists proof paperweights. I wonder what has happened to them. The Caithness exhibition of their paperweights was supposed to contain number one of each series, but over the years some of those first editions have crept out into the hands of collectors. They were the first of each series; they were not artists proofs.
Paperweights that are "one-of-a-kind" or marked 1/1 are something different again. Sometimes an artist is asked to produce a special paperweight as a commission, sometimes an artist makes a one-off paperweight as a gift or to celebrate something special, and sometimes they just decide not to make any more. These are not artists proofs because they were not the model for a series or for later production.
An experimental design is something different again. It may or may not have led to a series of similar weights being made. But the implication is that it did not. It was just an experiment.
Paperweights sometimes turn up marked "Sample". One US dealer assumed that this meant they were artists proofs, but that is unlikely. A sample would normally be produced to show the kind of work available, and there is no reason to think there wouldn't be dozens or even hundreds of such samples produced.
And finally, a word about "seconds". Some artists choose to destroy any piece that they do not think is up to their normal standard (Paul Ysart became famous for this). Some artists sell such pieces without their signature, refusing to sign anything that is not up to standard. And some outfits, like Caithness Glass, identify the second quality output with an indelible mark and sell it at a discount. I am not absolutely certain that current "seconds" sold by Caithness Glass are all so clearly marked as they used to be. But in every case, the item sold is still a piece of art made by an artist which may appeal to some people for the very features that caused its rejection.
There is a difference between a "second" which is essentially a piece which does not measure up to standard, and a lesser quality item which was produced for a different market.
Many artists employ a different form of signature for their best pieces to distinguish them from more run-of-the-mill output. And many European artists have different grades of work. Their very top pieces are often referred to as exhibition pieces; then there are collectors' pieces; possibly a grade below that for high class gifts; and then a range of designs intended for sale in tourist outlets and gift shops. The time spent by the artist will vary with the intended designation, and the quality and price will reflect this. It is up to the buyer to judge the quality of the item they are offered and to decide if it is worth the price asked. And most dealers will accept return of a paperweight that the buyer is unhappy with.