Any legacy .NET projects are made with .NET Framework 4.x (4.8.1 8s the latest). So if it's 4.x, or called .NET Framework instead of just .NET, it's referring to the old one.
.NET Core is no longer used as a name, and hasn't been since .NET Core 3.1. They skipped .NET Core t completely (to avoid confusion with the old one vut I think they caused confusion with this decision instead) and dropped the Core for .NET 5. Some people will still call .NET 5+ .NET Core (including several of my coworkers) which I'm sure doesn't help matters.
Mono isn't 100% completely dead yet, but you'll have little if any reason to use it (directly). I think the Mono Common Language Runtime is still used behind the scenes by the newer .NET when publishing on platforms that don't allow JIT (like iOS). They've started adding AOT compilation options in the newest versions of .NET so I expect Mono will be dropped completely at some point. Unless you want to run C# on platforms like BSD or Solaris or other exotic options that Mono supports but the newer .NET doesn't.
These days you just add the Microsoft package repo for your distro and then do `apt install dotnet-sdk-9.0` or whatever.
It's also been spreading into the official distro repos. Nix, Arch, and Homebrew all have it.