What is the mark of punctuation called that sometimes separates sections of a chapter with a glyph placed in the middle between the margins?
It's hard to explain, but here's a picture. Does this have a name? (Yes, I know this is a asterisk. Sometimes it's three of them, sometimes it's a stylized graphic. I'm looking for the general name of the...methodology/practice (?) of doing this.)
As explained by this page on Wikipedia, it's called a section break. I point you to the text of the annotation on the right side of the page where it pairs this term with the same kind of glyph you posted:
Open pages of the book Alice's Adventures in Wonderland, showing an ornate section break on the lower left page created from asterisks. It is used to signal a pause for the reader and a transition in the narrative.
And also the section on flourished section breaks:
Space between paragraphs in a section break is sometimes accompanied by an asterism (either proper ⁂ or manual * * *), a horizontal rule, fleurons, or other ornamental symbols. An ornamental symbol used as section break does not have a generally accepted name. Such a typographic device can be specifically referred to as dinkus, space break symbol, paragraph separator, paragraph divider, horizontal divider, thought break, or as an instance of filigree or flourish. Ornamental section breaks can be created using glyphs, rows of lozenges, dingbats, or other miscellaneous symbols. Fonts such as Webdings and Wingdings include many such glyphs.
In HTML, horizontal rules can be generated using the hr tag, which generates a paragraph-level thematic break. For more ornate presentation, CSS can be used to replace the line with an image.
As you've rightly acknowledged, the form a section break takes is entirely up to the person creating the publication.
Broader terms for glyphs used to separate text and do other non-text jobs are dingbat and ornament or printer's ornament. More info at the Wikipedia page for dingbat.
According to that same page, the specific character you're asking about is Unicode character U+273D, described as Heavy teardrop spoked asterisk, or possibly U+273B, Teardrop spoked asterisk.
*===,=*==,==*=,===*separating 5 chapters; but I've only seen it as an appropriate graphic, not as text.