What is an internal search engine?
An internal search engine is search over a company's own private content rather than the public web. It indexes the documents, wikis, chat threads, tickets, code and mail that employees produce, and makes them findable from one place. The term is used interchangeably with corporate search engine, internal search software and internal enterprise search, and in practice they all describe the same product category.
Two properties separate it from anything you would put on a public website. First, coverage crosses system boundaries: the tool has to authenticate into applications it does not own and keep an index of them in sync. Second, it is access-controlled per person, so the same query returns different results for different employees depending on what each is entitled to open. A public search engine has no equivalent of either problem.
The confusion worth clearing up is with site search. A search box on your own website is also technically internal search, and it is a completely different product with different vendors, because everything it indexes is public and permission handling does not exist. If a comparison article mixes the two, it is not written for the buyer you are.