As I indicated, this is not something I know a lot about, so I am hoping someone else will be able to help.
But I have to say that the whole thing feels wrong to me. Specifically, the "second order" concept relates to provable
bounds on the error, not necessarily to the actual error; and if the actual value could be estimated accurately by making the assumption they do, that would be taught as a standard method, which it is not to my knowledge.
The pages you attached do not quite apply to the differences that you have in your problem, but just to the actual error. But when I searched for information about the topic, and included the phrase "error multiplier", the first hit was apparently the very notes you are using, which suggests to me that this may be unique to them:
https://mei.org.uk/files/conference07/B7.pdf (When something is only taught in one place, I worry.)
The next pages after what you showed do deal with differences, and evidently are the basis of your work. And their reasoning parallels what I had worked out.
But your actual question is not so much about the method, as about the very claim that the ratio of differences should be 0.25, when your data show about 0.0025. If I had more experience with these error differences, I would probably have more to say; but at this point, I just have to question whether the data you were given are described correctly. Are they really the midpoint approximations of some reasonable integral for n=16, 32, and 64?
