Waterfall is a caricature straw man process where you can never ever go back to the drawing board and change the requirements or specifications. The defining characteristic is the part where design up front, you can never go back and really really have to do everything in strict order for the whole of the project.

Just having requirements and a specification isn't necessarily waterfall. Almost all agile processes at least have requirements, the more formal ones also do have specifications. You just do it more than once in a project, like once per sprint, story or whatever.

Waterfall certainly has processes for going back and adjusting previous steps after learning things later in the process. The design was updated if something didn’t work out during implementation, and of course implementation was changed after errors was found during testing.

Now that agile practitioners have learned that requirements and upfront design actually is helpful, the only difference seems to be that the loops are tighter. That might not have been possible earlier without proper version control, without automated tests, and the software being delivered on solid media. A tight feedback loop is harder when someone has to travel to your customer and sit down at their machines to do any updates.