>> ASP.NET Core in .NET 6. I've been on .net since 1.1 and .net Core since 1.0 and this statement makes me so confused. Core lives _inside_ of .net 6? Is there a non-core ASP.NET that also lives inside that framework? I thought .net 5 was where the legacy 4.x branch and core 3.1 was married in holy matrimony, minus the worst of the legacy garbage.
There is .Net Core, which was a rewrite of the .Net runtime to make it platform independent.
Then there is ASP.Net Core is which is a rewrite of the ASP.Net web stack. Despite the name, this was not initially dependent on .Net Core platform, it could run on both .Net Framework and .Net Core.
ASP.Net Core is the only web framework running on .Net Core though, the old ASP.Net framework is not ported to .Net Core. So to answer your question, no, there is not a non-core asp.net running on the .Net Core platform.
Then there is EF Core which was a rewrite of Entity Framework. Again, this was not initially dependent on .Net Core, although later versions is.
Basically "Core" just means "new backwards-incompatible rewrite".