Now, this first file the browser requests is a html file. The html file is parsed and the browser decides how to render its content. So one of the important concepts to retain here is that the html file is consumed by a parser.
The web server is a software that listens on a port and processes requests for files. If the file is a static file (we mean static here in the sense that the file is already created) then it is just copied as-is to the requesting client. If the file is dynamic, meaning that it is created each time the file is requested, the web server requests the file to be generated by the software that generates the file (either a running process, a loaded library, or spawning a process) and that software generates the file and sends it to the webserver, who, in turn, sends it to the client.
Now, how does this affect you? If any file on the server is dynamically generated, then there is software running on the server that tells it how to generate the file. The people who program these software are considered "server-side" programmers.
Many web frameworks, and dare I say, methodologies (MVC et al) have been developed to create a boundary between the pure "server-side" work, and the "client-side" work.
I forgot to mention, oh dear, the data people. The data storage people are even more server-side than the server-side html-file generating software writers. The data storage, whether relational database, NoSQL, or otherwise, are another thing altogether. I mention this because the Big-Vendor-touted frameworks and methodologies (MVC et al again) seemingly make it easy to just "simply bold that on".
Woah, what a longish answer.
I make this seemingly rambly answer to challenge your statement that there are server-side developers and client-side developers. If you deliver your information system through a web site, everything has to be stored, organized, and managed on the server. And it's a big mess, and unless you really learn how it all works, you're going to have a hell of a time making it work well. So it's all server-side.