Jean,
First, I am not sure what the origin of the terminology is but it is, at best, misleading. Premium typically refers to value/price above expected/required level.  Thus, you have to clarify that your terminology and advice apply only to new gTLDs.

Second, your post boils down to saying that they are whatever the registrar decides to call premium domain names based on, presumably, the ability of the registry to sell them at a price higher than their market value. If a registry wants to sell them at their fair market values, they would auction them off. Obviously, unless registries are irrational, they would not sell them for less than their market value.  Thus, the premium designation for the new gTLDs must be a marketing gimmick.

Third, you describe the concept as often involving “a memorable generic name, short or medium in length and highly valuable due to its relevance in search.” For one, this is not an actionable definition. Memorable, per se, is not easy to measurable. Moreover, the length being “short or medium’ is not well defined. In addition you describe them as also being “highly valuable.” But isn’t their value driven by their desirable attributes such as those you start the sentence with? Thus, high value is a result of having favorable attributes for branding and signaling, which you forgot to mention. What we need are statistical models with quantifiable descriptors that tell us what drives value in the eyes of the market, which are similar to those that have been used to price domain names.

1) Premium domain names for new gTLD = get there first to acquire a fantastic domain name.

2) A Premium sells at a higher value because it is explicit and a perfect match to what someone searches for. Re-selling a Premium domain name also allows to earn a lot of money. This is called “domaining” (may I suggest to read domaining.pro and domaining.com for thoses interested)
What is great and special is often more expensive: why do you think .COM generic domain names sold for millions of dollars? The same applies for new domain names. A fair market value is decided by the seller : a Registry is alone on his market so - of course - he decides about the price! Regarding what you call a marketing gimmick: there is an offer = take it or leave it for another domain name. No one forces anybody to acquire a Premium domain name.

3) Thank you for adding to my definition that high value is a result of having favorable attributes for branding and signaling. But you know I wouldn’t focus too long on waiting for statistical models because good Premium domain names don’t remain on the market very long and when it is too late…it is too late. Expect to pay twice and up to 100 times the price of this Premium if you missed it.

(1) So, why don’t people snap them up quickly? Do we need high-frequency trading software? Or are these “fantastic” profits illusionary?

(2) You are trying to demystify a concept. Yet, you say that a “Premium sells at a higher value” but still do not specify higher than what. You add that such domains “also allows to earn a lot of money,” a statement that underscores a lack of understanding of basic financial concepts. So, unfortunately, the concept is still a mystery.  To demystify the concept you need to focus on return, not price. For example, I can sell you a small store that allows you to earn tons of money, but that does not necessarily mean it is a good investment.  For the store to be value adding, its selling price must be less than the value of all future cash flows generated by the store, i.e., expected return on the store should more than compensates for the inherent investment risk. This concept is introduced in a first undergraduate course in finance. Moreover, you have caused tremendous damage to the concept of domaining and the industry, in general. You have defined domaining as selling domains at a price higher than their worth. Nevertheless, I agree that some domain names can be worth millions, but you have not explained why. Furthermore, you even use your own definition of a fair price.

(3) You say that “good Premium domain names don’t remain on the market very long.” So, now there are “good” and bad premium names? What market are you referring to? Moreover, you are in effect saying that if you pay $10 for a perishable item that is worth $10, you would not be willing to sell it for $12? I guess not; because I should “[e]xpect to pay twice and up to 100 times the price,” i.e. expect to repurchase it at possibly $1,000. I didn’t realize that registries are so irrational that they are willing to sell them at 100th of their worth!

Good article but you should of mentioned the restricted domain names that the registry deems to be ones they can not sell at this time. There is also the fact of some registrars are not filtering out the premium domains so the buy tries to buy a premium domain to find out they cant through that registrar.

You are right, I forgot to mention them but I tried to make the example simple. Thanks for adding.