Some people are interested in only one of these, so the distinction is not pointless.
Distinction springs from the person themselves.
I think ClumsyPilot's way to put it was just fine.
I just said that the distinction is not "pointless" or "gatekeeping", as some people accused, but pretty common convention which can be quite useful for readers. In particular, you are pushing back against something nobody said.
My "media server" (browser and downloaded media played via smplayer on a stock Ubuntu install) emerged from an experimental server running a lightweight distro that I used to do anything and everything from. Once I found that which parts of the media usecase fit into my partner and my lifestyle, it graduated to a stable decently-specced Ubuntu machine that is rarely touched other than for updates and downloading new content.
It looks to me that rolling your own media center is something to experiment with and learn something from.
Not to mention even if it was just a Plex server, that still meets your criteria.
It grew over time to serve many purposes, has seen many stopped/failed attempts, has had many lives (I mean, recreate from scratch to learn X, Y or Z).
There's not a particular day or a particular addition or a particular level of complexity when it became a homelab, I think of it as such from day 1.
Here are a few of the deployments that originated in my home lab but ended up in a prod environment at work
1. Replacing hardware load balancers with haproxy. (This started with a few options, including Nginx, and some others. But haproxy and it’s web management, csv monitoring gave me the best capability to integrate at work)
2. Vpn appliances for covid. I was able to whip up 2-4 scalable VPN appliances based on openvpn in “1/2 a day” at work because I was able to flesh out most of that at home.
3. Vulnerability scanners
4. HIDS security tools. In the end we went with a OTS vendor but options like ossec, wazuh etc were ruled out in a lab.
5. Ansible (over some of the other options)
6. Squid for reporting on the HIDS mentioned above.
There’s probably more. I know there is. And point blank a lot of this stuff had mock ups done at home because I have full control and am not subject to auditors etc when evaluating them. Whereas I do thst at work and I have to do more work writing up justification or change requests etc. it’s just easier at home.
All that said I try to keep the house as flat and plain Jane as I can.
I also have a lab hypervisor (Dell T30). This is where I feel free to experiment with VMs and accidentally on purpose have to rebuild it every once in a while, take it down to swap out hardware, etc.
My "media server" consists of a web application, backend application, multiple *arr services, transcoding automations, fibre termination, user account management shared across multiple machines and services, multiple VLANs and LUNs, etc.
All these are spready across 16RU or so, but really only serve as a "media server".
One of the most irritating things about "homelabs" is that most people seem to think a "homelab" means "a rack of very expensive, way-overspec'd ubiquiti gear, an OTS NAS unit, and a docker container server running media server/torrent shit."
I have a laptop running a dozen different containers - bookstack, torrent client, rss reader, and so on. I don't think of it as a "homelab."