What does it mean that a protocol realizes "strong mutual authentication"? What does it mean that a protocol realizes "strong mutual authentication"? And What does it mean that a protocol realizes "weak mutual authentication"? Can you define them formally ? 

I have found a definition for "strong mutual authentication" (not formal):


  Strong mutual authentication requires not only matching conversation records: all principals’ records of the content and the order of all messages must coincide, but also matching views of the intent: all principals’ views of the purported sources and the intended destinations of all messages should also coincide.
Strong with regards to cryptographic protocol usually means that the protocol is secure if the underlying primitives can be proved cryptographically strong. In your case, that could be because authentication is performed using a symmetric cipher and a key that is only distributed for 2 participants.

But as indicated in the comments, "strong" depends on context, you won't get any assurances from it by itself.