.NET Core = cross platform software development platform. Includes a VM, compiler, tons of libraries.
ASP.NET Core = HTTP server + server side .NET libraries.
There is no ASP.NET Foundation, it's all part of the Dotnet Foundation.
Xamarin = .NET libraries for mobile development. Wraps Android/iOS native libraries.
Visual Studio = native Windows IDE for .NET and other languages.
Visual Studio Code = portable, Electron/HTML 5 IDE for Javascript/Typescript and other languages.
You probably want Visual Studio Code for HTML/PHP. But you should check out .NET Core, it can replace PHP.
By the way, if it works, it works, but I find it better to research and do at the same time. Downloading only after you've understood the org chart seems a bit too radical for me :)
.NET Core is as the name says, just the "core" of the .NET Framework, the part that's cross platform. However, I'm not sure that at this point the code's common, I believe at least a part of it has been reimplemented. I think that in the future .NET Framework will be based on .NET Core. Therefore .NET Framework will be .NET Core + Windows specific bits.
WPF is the fancy name for a .NET UI toolkit for Windows, basically. Think of it as a Windows-only GTK.
I tried Wikipedia, and Microsoft's own homepages for each, and am even more confused.
I'm a hierarchical thinker, and a hierarchical tree explaining the above would be very very helpful.
Disclaimer: I haven't downloaded any of these yet. I usually don't until I've understood something.
Bonus question: If I want to develop websites and electron apps using: HTML, CSS, Javascript and PHP, what's the minimum set of technologies (or whatever it is that Microsoft is calling them these days) I will need?