What happens if I simply add an iterator mechanism to HTML (well, I guess we need variables too)? Is it no longer a markup language here (I won't add anything else):
<for i=0; i<1; i++>
<html>
</html>
</for>
Better question, why don't we upgrade XML to do that?
I dunno, you're being pedantic :) Yes yes, the name clearly ends up "Markup Language" so yeah, with a very strict definition of programming languages, HTML is not one of them.
But if we use a broader definition, basically "a formal language that specifies behavior a machine must execute", then HTML is indeed a programming language.
HTML is not only about annotating documents or formatting, it can do things you expect from a "normal" programming language too, for example, you can do constraints validation:
<input name="token" required pattern="[A-Z]{3}-\d{4}" title="Must match ABC-1234 (3 uppercase letters, hyphen, 4 digits)" placeholder="ABC-1234">
That's neither annotating, just a "document" or just formatting. Another example is using <details> + <summary> and you have users mutating state that reveals different branches in the page, all just using HTML and nothing else.
In the end, I agree with you, HTML ultimately is a markup language, but it's deceiving, because it does more than just markup.
I think that it is a debate, and it depends on the role of HTML in your system.
If all you're doing is using HTML to "annotate a document with its structure and its formatting", then yes, I'll accept that it's not quite programming, but I've not seen this approach of starting with a plain non-html document and marking it up by hand done in probably over two decades. I do still occasionally see it done for marking up blog posts or documentation into markdown and then generating html from it, but even that's a minuscule part of what HTML is used for these days.
Your mileage my vary, but what I and people around me typically do is work on hundreds/thousands of loosely coupled small snippets of HTML used within e.g. React JSX, or Django/Jinja templates or htmx endpoints, in order to dynamically control data and state in a large program. In this sense, while the html itself doesn't have control flow, it is an integral part of control flow in the larger system, and it's extremely likely that I'll break something in the functionality if I carelessly change an element's type or attribute value. In this sense, I'm not putting on a different hat when I'm working on the html, but just working on a different part of the program.