Don't forget that ASP.NET Core can target not only .NET Core but also .NET Framework. We had to target .NET Framework in our latest ASP.NET Core web app because we still need access to COM+/Interop.
This is exactly the issue being discussed here - ASP.NET Core will not run on .NET Framework in the future.
Today .NET Core is younger and smaller so it's easy to also run .NET Core apps on .NET Framework since .NET Framework is larger and has everything implemented already - but as we progress with new APIs in .NET Standard, .NET Framework will continue to fall behind (until it gets those major multi-year updates). So you can still use older ASP.NET Core versions (like today's 1.x) on .NET Framework but eventually the newer versions will be too far ahead of the .NET Framework.
However, .NET Core is cross-platform and runs just fine on windows, brings its own dependencies, and will be able to use .NET Framework libraries so this whole issue is overblown. System.Drawing, ActiveDirectory, and other APIs will be added to .NET Core soon enough. If you really need COM interop, then either just stick with .NET Framework apps or use older .NET Core apps.
".NET Framework" made sense when there was only one real framework available, but there is now Core, Compact, Mono, Unity, etc. which makes it confusing.
".NET Desktop" can be used instead, although it's also misleading since it has nothing to do with desktops. ".NET Full Framework" is also used often.
Perhaps the best name would be ".NET Legacy Framework" with the understanding that it is a heavy, slow-moving framework implementation that only runs on Windows (and is limited by the Windows version as well).