First off, I want to note that these sorts of demarcation problems only very rarely admit of anything like a generally acceptable solution. You can always find borderline cases and people willing to quibble over them. The best known example of this is probably defining what gets to count as art and what doesn't. Roger Ebert famously argued that video games weren't art, many people disagreed with him, that sort of thing. Another good example is science. Is astrology not science at all or just a really bad scientific theory that has been thoroughly falsified? Well, that depends on what characteristics you think something must have to qualify as science, which turns out to be pretty hard to spell out in sufficient detail.

Now I mention all that just to say that getting a clear answer here that everyone will happily accept is probably just not a realistic goal. Therefore, the best practice is really just to be super clear about exactly what you understand by the relevant terms. In this sense, Gordon has done a fantastic job here; he knows that people might understand these terms differently than he does, so he clearly and explicitly let's you know what sorts of things he means to pick out when he uses the terms.

Now, something that does seem really plausible, to me at least, about the way he is carving up everything here is that he has defined the history of ideas as a subset of intellectual history. This seems right to me. Intellectual history ought to be the broader category, since the literal words seem to purport that more than just the ideas themselves will be at issue. History of ideas, by contrast, does seem naturally to fall into the category of intellectual history but to not be quite so broad.

Let's take philosophy as an example. Actually, let's be even more specific and just take the early moderns. A history of ideas of the period would, of course, involve discussions of the relevant arguments and concepts at play. So, for example, considering the skeptical arguments of Descartes and Hume, the dispute about the nature and origins of ideas, shifting understandings of causation, that sort of thing.

But importantly, at least as I understand the terms (which I think is also how Gordon is understanding them), a history of ideas would not involve stuff like, say, the biographies of the thinkers, the establishment of prominent intellectual centers, or the impact of those ideas on the broader culture, etc. All that stuff would fall under intellectual history.

A really interesting way of driving this home is the different ways that disciplines approach certain texts. If you take a class on St. Augustine's Confessions from a history department, you will talk about all sorts of stuff. The relationship of Augustine to his mother, broader Imperial society, Augustine's occasionally poignant reflections on his youth and those around him, etc. But if you were to study the exact same text in a history of philosophy class, none of that would come up, except maybe in passing. Instead, the entire focus would be on the arguments: What is the problem of evil, how does Augustine respond to it, is that response plausible both in the context of Augustine's philosophical system and if we disagree with that system does it fall apart, what the hell actually is Augustine's position on the nature of time, etc. You would touch on all this stuff in the history department course as well, but in a philosophy classroom it would be all that you talk about. I seriously took a course in grad school in a philosophy department where about half the class was on Confessions and Augustine's mother never once came up in lecture or discussion, despite the fact that she really hangs over the entire book. But that just wasn't the part of the text we were studying.

What is going on here, I think, is that historians of philosophy tend to be zeroed in on history of ideas to the exclusion of everything else, whereas a history class tend to cast a wider net. Neither approach is better or worse in any objective sense, it is just that they are doing different things.

All of this was just a rather long winded way of saying that I don't take much issue at all with Gordon's way of cutting reality at the joints, as the saying goes. In fact, I rather agree with him. But that said, I've also heard people use "intellectual history" and "history of ideas" interchangeably before, especially in conversation. Which is also fine. The really super important thing is that, whenever you are using terms like these with no obvious definition, that you take the time to clearly define exactly how you are using them.

Thank you so much for taking the time to reply! Really interesting to get your take on the matter, and this has definitely helped me get my head around both applying and understanding the use of such concepts :)