Almost there.

The CLI is the standard, the CLR MS' implementation, you've got that right. However, the Base Class Library is part of the CLI. It is the absolute most basic set of libraries that every conforming implementation of the CLI must include. More specifically, every conforming implementation must implement what the standard calls the "kernel profile" (section IV.3.1) which includes the BCL (section IV.5.3) and the Runtime Infrastructure library (section IV.5.2). The CLI standard also defines several other libraries (section IV.5) which are optional. The complete contents of these libraries are defined not in the standard text but in the XML specification enclosed with it.
The .Net Framework Class Library is a superset of the libraries defined in the CLI (including the BCL); it includes those libraries as well as several that are not standardised, such as System.Windows.Forms and System.Web.
Mono has in their work of duplicating .Net been very careful to separate that which is standardised and that which is not, so that in case they receive a cease & decist on those portions of .Net that aren't standardised they can at least continue with those portions that are without any effort of removing the non-standard bits.