From a game-theoretical perspective there is no ambiguity in the rules: checkmate and stalemate are both won positions, whether the game ends there or whether one in some cases would have to play one more move to actually capture the king.

This isn't correct because players aren't perfect agents. They do not always identify and make the most optimal move. It is possible that someone in "checkmate" will move their king into an easily missed "check" (e.g. a check from the coordinator), and the opponent won't realize they could still capture the king.

Whether this is allowed or not clearly makes a difference when played in any setting with a third-party overseer (an arbiter, a computer client, etc).

he actually changed his mind at some point how the game should be played

Well, we know this is the case to some degree because the 1968 rules are very different from the 1963 rules. He then decided those had been a bad idea, and went back to the 1963 rules.

He even stated in both 1962 and 1963 that the game was still freshly developed and may still have issues. It isn't that surprising for some language of a newly designed game to be contradictory; an aspect of the game started out one way, and later in the design process was changed, but the rest of the rules weren't updated accordingly to reflect the change.

It should be assumed the part of the contradiction that was intended was the part the creator later explained was intended (capturing the king), especially when he also said the other part of the contradiction was not intended (the checkmate).

E.g. an on-line interface would be likely to make it impossible to enter moves that expose the king to begin with, and perhaps automatically resign for a player that is checkmated.

This reminded me that Abbott promoted "Zillions of Games" as a software that accurately enforces the rules of Ultima.

I just found and installed this piece of software and tried a game of Ultima on it. You can play against the computer or against other people.

In this interface that Abbott promoted as correct, you are allowed to enter moves that expose the king. You are allowed to move your king from a safe position into an attacked position. It does not automatically resign, allowing the opponent to make a mistake and miss a capture. You must capture the king to end the game, whereupon it says "King captured. Black wins!"

what is the proper procedure for claiming the victory when the opponent exposes his king to capture

According to the software Abbott promoted, the proper procedure is to capture the king.

I would not exclude that the sentence was just intended to convey that stalemating is an alternative.

No. As someone who loves Robert Abbott's work, I'm not okay with this blatant misrepresentation of his words. His paragraph again was:

The first puzzle is Mate in 1 and all the others are Mate in 2. This follows the conventions of chess problems, even though in Ultima the object is to capture the king, not achieve check mate. So, Mate in 1 should be translated to Capture the king in 2 moves (that is: White moves, Black moves, then White moves and captures the king). Mate in 2 translates to Capture the king in 3 moves.

After saying "in Ultima the object is to capture the king, not achieve check mate," he elaborated with "So" and even explained what he meant with "(that is: White moves, Black moves, then White moves and captures the king)." There is no way to misunderstand this language, and the attempt to do so makes me question the motivation behind it.

And again, these puzzles which require the king to be manually captured per the rules of Ultima were made in 1964, and explained in 2004. It makes it very clear what Abbott's intention was, and the only interface Abbott promoted for playing Ultima works that way as well.