At that point the computers is promoted (or enslaved depending on pov) and called a (home) server.

One nuance. You don't need to provide access from outside your home network if you don't want to open the ports, i prefer that most of services remain local for security anyway. Once those ports are open, you or anyone else can connect to the services using your_external_ip_address:portnumber

You also dont need to run it 24/7, i started with a minecraft server that ran on planned times only, then a system that was online during afternoon-evening but not at night-morning.

It is also possible if not recommended, to use a vpn to tunnel in your network and than you can access everything just like if you where at home from anywhere.

If it all sounds intimidating, a laptop to try things out will do fine. Don't expect a perfect polished system on first try, i have redesigned my network from scratch multiple times over the years. If you do want to get serious there are plenty of offices that have "old" desktop servers with pretty impressive hardware, the fans may make a lot of noise though.

The Hardware in my current home server is mostly parts that got upgraded in my main pc. I have reached a point where i am willing to sometimes spend extra money on what my partner sometimes dubs "my second pc" but it is also doing a lot for us at this point. Much cheaper then a store-bought NAS.