As hinted in comment by @Ron Maupin, there isn't an universal denomination, this will be vendor dependent, so:
Are there different implementations of it?
Yes, almost one different per vendor, however most of them are very similar.
Examples:
In Cisco world you have basically:
- user exec mode
- privileged mode
- configuration mode
In Mellanox:
- Standard mode
- Enable mode
- Config mode
Incidentally "Enable" is the command in Cisco to go in privileged mode, we can clearly see the heritage.
In Juniper:
- Operational Mode
- Configuration mode
In Fortigate, while you don't really have mode you have a command tree where the branch which allow configuration change is named... ....configure.
You can see that in most case, the name of the mode allowing to modify the configuration is called either "config" or "configure" or "configuration", while the other mode(s) have different names.
However if you speak about "operation mode" or "user mode" as opposite to "configuration mode", any network engineer should get it.
Is it scoped to *nix-only network config?
No. Cisco IOS is not *nix (while Cisco NX-OS is), and it is arguably the first major Networking Operating System.
Are there more modern variants, like git vs svn?
This is comparing apple to oranges, this question doesn't really have a sense in the world of networking, and also git is not a modern version of SVN, those are two different Version Control System, even if svn has become less popular.