This can be confusing at first, but it’s important to remember: at the end of the day, “ASP.NET Core” is just a bunch of NuGet packages that can be installed in your project.
ASP.NET Core on .NET Core is cross-platform ASP.NET Core. It can run on Windows, Mac, and Linux (including Docker). The server doesn’t need .NET Core installed – the dependencies can be bundled with the application.
ASP.NET Core on .NET Framework is ASP.NET Core on the “full” or “desktop” .NET Framework (e.g. .NET Framework 4.6.2). These applications can only run on Windows, but everything else about ASP.NET Core behaves the same way.
According to the benchmarks you linked, both will have higher performance than ASP.NET 4.6, although .NET Core is currently the highest:
ASP.NET 4.6: <50k req/sec
ASP.NET Core (CLR): 400k req/sec
ASP.NET Core (.NET Core, Linux): 900k req/sec
ASP.NET Core (.NET Core, Windows): >1.1m req/sec
However, these benchmarks are slightly older (February 2016) and reflect pre-RTM code. I wouldn’t be surprised if they’ve improved since then.