Thats routing in the networking world. But of course you can call routers servers by your definitions. They are computers in their own right serving up data to other routers or to the final PC.
Its a very circular definitions you use. And you are welcome to use them. But do not expect the networking world to adopt those definitions or anyone else for that matter. A server in the world of networking is much more than a device that can spit out data upon request, and you’d be laughed at by network people for suggesting such.