No, both of them mean that Janet is doing the thinking. Only in directly quoted speech do personal pronouns retain their original referent in English: in reported speech, they are substituted by pronouns relative to current speaker.
John said "They were looking at me."
means that they were looking at John, but
John said (that) they were looking at me.
means that they were looking at whoever said this, not at John.
To get the second meaning, you cannot have "me"/"my" - but "don't bet on you(r) thinking" would be odd: I would say something like "don't rely on thinking", though probably I would avoid the gerund entirely "Don't bet that you can" or something like that.
The difference between your two examples is stylistic. Old fashioned prescriptive grammar taught a bizarre rule that "don't bet on me thinking" was ungrammatical and you were supposed to say "don't bet on my thinking", but the former has been in common use since before this rule was invented. See Gerund with genitive.
Edit: with the comma after "me", the "thinking" becomes a participle, and could indeed refer either to the subject (Mike) or the object ("me", i.e. Janet). I missed that reading.