I don’t think any of these examples were what Illyes meant when he talked about HTML structure. I believe he was referencing the common arguments of heading nesting, bold tags, etc. TL;DR ...

Keep the content structure in mind. You don’t start your page with an H3 heading. Instead, it should H1 heading followed by an H2 heading followed by a few H3 headings. There can be several H2 and H3 ...

H1 – H6 elements “briefly describe the topic of the section they introduce“.They form a page HTML semantic structure that can “be used by user agents, for example, to construct a table of ...

Thus, if the site uses invalid HTML markup (e.g. multiple H1 tags, or one and the same headings on each page), the worst what must happen is that Google will just disregard the overall structure ...

Lower degree headings should be contained within headings of the next highest degree (i.e., one should not skip heading levels, such as from a Heading 2 to a Heading 4, going down the page or document ...