I think the confusion is about what 'run' means in a web-context. The code you want your clients to run, the javascript you are writing as your frontend, the buttons, the outgoing requests and everything, runs in the users browser. That browser gets the application/page from a server. This server for your frontend is technologically the same as the server for your backend, it is a process running on a machine (that is typically not the users machine) responding to http-requests from the internet.

Nowadays, a common way to write web-applications is to have two servers: A 'backend' server that only deals with data, not html, and a 'frontend' server that serves the web application itself. The frontend is typically architected in such a way that you don't need to think about the server, only the browser. So people will use the word 'server' to mean 'backend-server'

There are of course other ways an application can be distributed. Data and UI can be handled by the same server. You may write a distributed application not involving browsers. You may have multiple servers for data. There are many options, I have only described one modern common architecture, because this architecture is so widespread that people will often assume it unless otherwise noted.

Does this answer your question?