If that's your impression, then I think you have been. By full-stack developer, many people in the start-up world simply mean someone who can work at all levels of web development, from server-side programming to Javascript, CSS, and HTML.

It's not an unreasonable use of the word, since web developers have long talked of "their stack", meaning what they use for server-side framework, caching, http server, as well as client-side frameworks and development environments.

I don't think most people use the term to include sysops, although in a small start-up you will inevitably be doing a lot of that.