But I think the general meaning of a cutoff date, D, is:

The model includes nothing AFTER date D

and not

The model includes everything ON OR BEFORE date D

Right? Definitionally, the model can't include anything that happened after training stopped.

That's correct. However, it is almost meaningless in practice as it might as well mean that, say, 99,99% of the content is 2 years old and older, and only 0,01 was trained just before that date. So if you need functionality that's dependent on new information, you have to test it for each particular component you need.

Unfortunately I work with new APIs all the time and the cutoff date is of no much use.