Too many semantics and statements based on opinion.
First off: C# isn't an interpreted language; the CLR and JVM are considered "runtimes" or "middleware", but the same name applies to things like Perl. This creates a lot of confusion among people concerned with names.
The term "Interpreter" referencing a runtime generally means existing code interprets some non-native code. There are two large paradigms: Parsing reads the raw source code and takes logical actions; bytecode execution first compiles the code to a non-native binary representation, which requires much fewer CPU cycles to interpret.
Java originally compiled to bytecode, then went through an interpreter; now, the JVM reads the bytecode and just-in-time compiles it to native code. CIL does the same: The CLR uses just-in-time compilation to native code.
Consider all the combinations of running source code, running bytecode, compiling to native, just-in-time compilation, running source code through a compiler to just-in-time native, and so forth. The semantics of whether a language is compiled or interpreted become meaningless.
As an example: many interpreted languages use just-in-time bytecode compilation. C# compiles to CIL, which JIT compiles to native; by contrast, Perl immediately compiles a script to a bytecode, and then runs this bytecode through an interpreter. You can only run a C# assembly in CIL bytecode format; you can only run a Perl script in raw source code format.
Just-in-time compilers also run a lot of external and internal instrumentation. The runtime tracks the execution of various functions, and then adjusts the code layout to optimize branches and code organization for its particular execution flow. That means JIT code can run faster than native-compiled code (like C++ typically is, or like C# run through IL2CPP), because the JIT adjusts its optimization strategy to the actual execution case of the code as it runs.
Welcome to the world of computer programming. We decided to make it extremely complicated, then attach non-descriptive names to everything. The purpose is to create flamewars over the definition of words which have no practical meaning.