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in Chess the object is to capture the king, not achieve checkmate.

No one says that of orthodox Chess, because it obviously would not be true there: there is no way to win an orthodox Chess game other than by checkmating the opponent. Not so in Ultima, where stalemate is a second winning condition, rather than a draw. And 'smothered stalemates', where a player doesn't have any moves even when he would be allowed to expose his king, are not nearly as hypothetical in Ultima as in orthodox Chess: Pieces without replacement capture can easily be boxed in, even if they are not frozen by an Immobilizer. So I would not exclude that the sentence was just intended to convey that stalemating is an alternative.

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. So the whole discussion is about what I would call "game etiquette": what is the proper procedure for claiming the victory when the opponent exposes his king to capture. Should you actually capture the King and press the clock, or should you just stop the clocks and submit the claim? Or is this not a victory at all if there was an alternative, and should you allow the takeback and that alternative to be played, possibly in combination with a time penalty?

Some of the quotations attributed to Abbott seem to contradict each other, though. There can be two reasons for that: he actually changed his mind at some point how the game should be played, or he has always thought the same about that, but failed to express himself unambiguously. In the latter case we could speculate on what he actually meant. If it is correct that Abbott at any time mentioned that it was not allowed to expose your own king to check, this strongly suggests that he thought a move that did should be handled as per FIDE rules: take back and play another one. For if it would be handled the Shogi way (instant loss), there wouldn't be much reason to mention it at all: the game is over anyway, either by claim or by king capture, unless the players don't notice it, in which case it doesn't matter what the rule is, as it isn't going to be invoked. So it is a matter of weighing the "not by checkmate" against "it is illegal to put your own king in check", I would say the latter of those is the least likely to have meant anything else as what it says.

I can add that no matter what Abbott actually meant, tournament organizers are likely to enforce their own game etiquette. 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.