What should i call a "Paragraph with a header"...?

Solution 1:

The paragraph itself is still just a paragraph in its own right.

However, the so-called header that introduces the text it's part of is, in general terms, a heading:

[Merriam-Webster]
1 a : something that forms or serves as a head especially : an inscription, headline, or title standing at the top or beginning (as of a letter or chapter)

What the heading does is delineate a section:

[Merriam-Webster]
2 : a distinct part or portion of something written (such as a chapter, law, or newspaper)

It might be ambiguous, but in that definition "such as" is referring back to "something written," not to "a distinct part or portion." So, for instance, the chapter would have sections, not be called a section.

In short, in this context, a section is composed of a heading (sometimes called a section heading), followed by text, which is normally composed of one or more paragraphs (although sometimes, for example, just bullet points or figures).

These paragraphs are just part of the section.