A web server is server software, or hardware dedicated to running said software, that can satisfy World Wide Web client requests. A web server can, in general, contain one or more websites. A web server processes incoming network requests over HTTP and several other related protocols.
As David Malan says that web server is a piece of software written by people.
So you can as well implement your own web server on your computer and host your web sites.
The flasks web server is an already implemented server by other group of people for the benefit of them or you.
You borrowed the implemented web server just to serve your assignment or project files because you didn't want to implement your own.
To answer your question.
web server can refer to the software that serves up website data, or the hardware that serves the data, or both.
So flask web-app is a software which can be used on your computer to perform its purposes, and its purpose is to serve files on your computer.
Apache, flask web-server etc. are examples of a web server. Apache is not a computer so it shouldn't run another web server.
The Apache HTTP Server, colloquially called Apache, is free and open-source cross-platform web server software, released under the terms of Apache License 2.0. Apache is developed and maintained by an open community of developers under the auspices of the Apache Software Foundation.
License: Apache License 2.0
Initial release date: April 1995
Written in: C, XML, C++.