Regarding the questions: What do you call those divisions of a book bigger than a paragraph but smaller than a chapter? Is not such a division best known as a PARICOPE from the Greek PARIKOPE or "a cutting around" as in the selection of a passage from the Bible in religious services?
No: most English speakers will fail to understand the ecclesiastical term of art pericope even if you spell it correctly, and section (sense 2) is in everyday use for that purpose.
Also passage can convey the concept:
(Literary & Literary Critical Terms) a section of a written work, speech, etc, esp one of moderate length.
Source:Collins English Dictionary
LaTeX divides a Book into Chapters, a Chapter into Sections, a Section into SubSections, a SubSection into SubSubSections.
I think the notions propounded by LaTeX are representative of fairly modern thinking.
I am not sure whether a pericope corresponds to a Section, a SubSection, or a SubSubSection.