I think it is hard to argue against "The object of the game is to capture the enemy king." This is a direct quote from Abbott. There's no way to misinterpret that sentence to mean the object is checkmate.

From what I see, both the oldest source and newest source from Abbott agree on this. Even if the 1968 edition (which Abbott is on record disavowing by the way) includes contradictory language, it still says "The object of the game is to capture this king." This statement can't be ignored.

These statements are very straightforward and impossible to misunderstand. Every source from Abbott is in agreement. In each source, the object of the game is explicitly stated to be "to capture the king."

I believe "The object of the game is to capture the enemy king" makes it clear what the object of the game is, and there is only one way to interpret it. Even if some of these sources include contradictory language that also states the exact opposite, not all of them do (i.e. the rules as explained by Abbott on his website state that the object is to capture the enemy king, and they also specifically state that there is no checkmate). The only object every source from Abbott includes and agrees on is "to capture the enemy king."

Also, if it "moves and captures in exactly the same fashion as the chess king," it could not move into check.

I feel this may be a bit of a stretch. This interpretation would contradict the object of the game that is explicitly stated in the same line. Rather than assume Abbott was thinking of checking behavior and castling when he said the king "moves and captures" as the chess king does, it would probably make more sense to read his statement literally: the king "moves [1 square in any direction] and captures [by displacement]" as the chess king does. Taking his sentence at face value has the added benefit of not contradicting his very next sentence in the paragraph as well.

Even if there was a source in the past where Abbott indicated "the object is to capture the king" and at the same time he said "the object is to checkmate the king," that only tells us one of those contradictory statements would have to be wrong. Given that Abbott wrote an in-depth explanation about the object of Ultima on his website, it seems clear which option he intended to go with.

I am looking at this page mentioned earlier, and Abbott is even more explicit here.

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.

Note that if checkmate was the object of Ultima, there would have been no reason to include a paragraph here at all. Attempting to find checkmate is how these types of puzzles work by default. Abbott went out of his way to write and include this paragraph about how the object of Ultima is not the same as chess, and to make it completely unambiguous. It's pretty clear he didn't want any further confusion about this.

Going through the content of Abbott's explanation, I'll start by highlighting the short answer:

in Ultima the object is to capture the king, not achieve check mate.

In this case, he not only said "to capture the king" is the object of the game (matching his other sources), but this time he even elaborated that achieving checkmate is not the object of the game. This explanation directly addresses the contradiction that was found in one of the older sources and leaves no room for misunderstanding.

However, in case he still hadn't been clear enough yet, Abbott went on to explain the actual turn sequence that ends a game of Ultima:

White moves, Black moves, then White moves and captures the king

Note that in Abbott's example of how an Ultima game ends, Black makes a legal move that leaves his king under attack. It is obvious from this that there is no checkmate, otherwise Black could not have moved. Abbott's detailed move-by-move explanation removes all ambiguity and clears up any potential confusion. It rules out any possibility that checkmate is the intended object of the game.

On the Chess Variants article, there is also a 4th source linked, the original 1962 Article introducing the game. This article was brought up earlier, but again, this source matches the rest of them, clearly stating "The object of the game is to capture this king." The rules include no mention of checkmate at all. All it says about checking is "The same rules for declaring check apply as in chess." Of course, there are no rules that require you to declare check in chess, so that's a moot point.

Additionally, I see on Abbott's site that he was really enthusiastic about David Howe's "extensive" and "especially useful" write-up of an Ultima game ending. I found his write-up.

These are the last three moves of the game in his write-up:

White Chameleon at e8 moves to g6

Black must move LL at g7 (or suicide a piece)

White Chameleon at g6 captures Black K

Note that in this write-up, White moves a Chameleon next to Black's King, which Black has no way to avoid. Black is then required to make another move. White then captures Black's King to end the game.

This is another specific example for how an Ultima game ends, with Black making a move that ends with his king under attack, and White then capturing Black's king.

Earlier it was mentioned that Abbott recommended the chessvariants page at one point, but he did so in the context of wanting to share admiration for David Howe's animated illustrations. It appeared to be a short respectful way to introduce the topic, giving a general compliment lacking any specificity. Based on the rest of the paragraph, Abbott makes it clear that the part of the site's explanation he was impressed with was the animated illustrations that explain the piece movements. Even if all the context is ignored in order to assume the compliment was well-researched and literal, saying something is the best option available doesn't suggest that it can't be improved upon, or that the speaker made a point of proofreading the site in the first place.

Additionally, shortly after that paragraph where he says chessvariants is a good site, he then links to his own page which he had just written himself, where he states multiple times that the object of Ultima is to capture the enemy king, and that the object of Ultima is not to checkmate, and he even explains move-by-move how an Ultima game ends in order to make the object of the game perfectly clear.

Actually... Abbott mentions multiple times, across several different pages, that his Ultima puzzles (which required you to capture the enemy king) were created in 1964. This means the 1963 rules, which these puzzles were utilizing, also intended for you to capture the enemy king.

With the Abbott of 1964 and the Abbott of 2004 being in agreement that the object of Ultima is to capture the king, the conclusion seems pretty straightforward.