In my career (over 30 years now), full-stack in the context of web development has always referred to an engineer who could setup the tech stack and application from scratch. They can manage the servers, administer the network (as it relates to permissions for the various web-related processes), implement developer environments, code the back-end, code the front-end, some basic design skills, experience with developing UX, and is deeply familiar with the network protocols being used.

Anyone claiming to be full-stack should be able to handle light dev-ops for a small to medium sized company for at least a few days without breaking things. They should be able to sit in any other generalist coder's chair for a week. They should be able to work closely with a designer to help develop the look and feel of the application and have an understanding of the domain tools and language. They should be able to work closely with a DBA to help develop the best database for the application. They should be able to debug complex app behaviour with their understanding of how the different sections work, even if they don't have explicit knowledge of specific areas of the application.

A full-stack "developer" may only refer to the coding and design+UX portions of that description, but should still be very familiar with how the server software and network works. A single framework may provide a full-stack development solution (RoR, for example), but a full-stack engineer needs to cover a lot more ground than just development.