I think NAS was originally distinct from a File Server because it was used to describe devices that let you access the data across the internet- like accessing storage at your house when you were away. The name "Network Attached Storage" meant that it inevitably supplanted "File Server", since a File Server is, literally, Storage attached to a network. And I guess it sounds cooler.
So anyway, what you are looking for is indeed called a NAS nowadays. I don't make the rules.
Now, of course, if you go searching for NAS stuff you'll get recommendations for those proprietary devices you mentioned. But, you can build your own or install software on a computer/server and make it a NAS/File Server.
By coincidence, I set up a file server on my network just last week. I wanted readily accessible mass storage that I could access through network shares anywhere on my LAN.
I used OpenMediaVault, which is a specialized Debian-derived Linux distro designed for a bunch of different functions, but which serves competently for the task. Basically you install it and then once installed and setup, you can connect to the system through a web browser on another system on the network, and do the configuration to set up shares.
In my case I went a bit further and used some plugin software called mergerfs (which also has an OpenMediaVault plugin) which allows the creation of a "virtual" disk that is the merged view of multiple disks, which is useful since you can then share that virtual disk folder, and when you copy files to the server it figures out which drive to put it on based on how you have it configured.