In manuscripts, a text between the chapter heading and before the first section might exist. Does this have a name?
This seems to be a good community to know the answer. Thanks in advance.
Text between the chapter title and the first section heading is usually called the "lead paragraph" or "lede". Despite the name, this can be one ore more paragraphs of text.
If the text is a quotation, it is called an epigraph. See this APA blog post on how to format epigraphs.
Other than that, most citation guides do not seem to recommend the insertion of a section between a chapter heading and the next subsection.
But I have not found any particular term for that. The descriptive terms listed in the comments seem plausible to me (e.g. "chapter introduction" or a "preface" to the chapter with multiple synonyms).
It's become quite common in academic books in the STEM fields to add such a section as a kind of abstract for the chapter. The reason for this is that the publishers are now selling online access to individual chapters of books through their web sites. When you search on Google Scholar, Web of Science, etc., you'll often get results which are book chapters- if you follow the link to one of these book chapters, you'll get access to the brief summary, but access to the full chapter will be behind a paywall. I've heard these called "chapter abstracts."