Is there a term for a section of an article which contains a quote or snippet from the main text?

Solution 1:

I think it is called pull quote:

In graphic design, a pull quote (also known as a lift-out pull quote) is a key phrase, quotation, or excerpt that has been pulled from an article and used as a page layout graphic element, serving to entice readers into the article or to highlight a key topic.

It is typically placed in a larger or distinctive typeface and on the same page. Pull quotes are often used in magazine and newspaper articles, annual reports, and brochures, as well as on the web. They can add visual interest to text-heavy pages with few images or illustrations.

(Wikipedia)

A technique traditional to print is the pull quote, a snippet of content that stands out from its surroundings to grab a reader’s attention as they scan pages.

(htmldog.com/techniques/pullquotes/)