Learn English – How is the word ‘but’ used in the famous quote ‘What is life but a series of inspired follies?’

It is definitely not common, and just as you suspected it is a play on the previous two examples in the list of "Travelers" and "Givers".

It doesn't quite perfectly follow a trend: Traveler, Giver, Artist -- because the first two use the English language -er suffix to suggest the "actioner" of a verb (an action), and "Artist" is not "Arter".

But this is the syntactical joke being made: A traveler is one who travels, as an artist is one who "arts".

But if the question is whether or not this is common practice in English, then no, it is definitely not.

The act of artistic expression is understood to be an act of creation or construction, to do or make or create something.

The common expressions are: an artist creates...; an artist makes...; an artist expresses...and other such phrase statements.

Whereas it is proper and common English practice to say: a traveler travels; a giver gives.

The only transitive verb usage I've heard in the context of chess itself is the one given as the second definition in your question. As implied by the definition, the object of the checkmate action must be the king piece of your opponent, as no other piece is used to define the state of checkmate. The object cannot be the opponent themselves (though there does seem to be an exception for formal third person usage - see the final paragraph below). There isn't any real restriction on the subject - either the player or the pieces could be said to be performing the checkmating.

In short, I don't think your first sentence is correct, but any of the other three examples you give seem reasonable. The second sentence seems correct through implication: by only denoting colors, you are implying that it is the black king which has been checkmated. The third and fourth are correct through explicit usage of the king as the object of the verb.

I will say, however, that I've heard this transitive-verb usage of the word checkmate only rarely, and would likely not use it myself, instead preferring to find ways to phrase my sentences such that checkmate appears only as a noun.

Note that this is true only for use within the game of chess itself, as specified by your question. In any other context, checkmate is frequently used as a metaphor, and many things can be checkmated, including people, armies, and objects. However, during an actual game of chess, using checkmate as a transitive verb essentially requires that the object be the king chess piece. Any other usage will sound very strange to a native speaker (though they may be able to determine what you mean). One would never say they had "checkmated your knight" or "checkmated your rook" in the game, and attempting to use such a phrase would likely confuse the listener. One could say "I have checkmated you" perhaps, during a game, but again it will sound fairly odd to a native speaker. Instead, they would say "I've put you in checkmate" or something similar. If they are bound and determined to use checkmate transitively, they may say "And now I have checkmated your king", but it will sound fairly formal and stilted, if not incorrect.

One of the in-game uses of the transitive verb form listed in the OED in a comment below was interesting, however, and does sound formal but acceptable in contemporary English: "A player must checkmate his opponent within fifty moves" and similar sentences do seem to work due to them being in third person. In contemporary English this will sound formal and somewhat "instructive", but will also be perfectly acceptable usage. However, attempting to use second or first person ("I checkmated you!" or "Will you be checkmating me soon?") will still sound odd in modern English.