Do you have an example? There is no unambiguous way to define "web server"! Consider all the ways you can rent a host in virtually every state between disassembled parts on a table (Newegg) to renting DC for your own hardware (Hetzner) to running an httpd process running in a hypervisor running on a fully managed server in a DC (VPS) to sharing wordpress db/virtual host space on a LAMP process (wordpress.com). And everything in between. (And that's not every option, either. "Push markdown and let github/gitlab publish it for me" or "write a gist" are in this space, as is codepen, meteor, aws, and reddit - really anything tool or stack that lets you "change the state of the public web". Even HN!).
In fact, understanding this spectrum of options is a task in itself - imagine starting from scratch and trying to pick a path!
At what point on this spectrum do you transition away from "self hosting"? Personally, I put the line at "vanilla VPS". I do not consider sharing wordpress space self-hosted, nor is having a webmin. (And yet, either is certainly more self-hosted than a Facebook group page!) My approach was "device oriented" in that I wanted to act as if I was configuring a fresh piece of server hardware with a new OS and so on. The cheapest way to get that is a VPS. It is at this level that I feel I have the most flexibility, the least lock in, and, importantly, the most leverage.