Dear Arso,
The rationale to the Lubavitcher’s mind is that term ‘The Rebbe’ means one who spreads Toras Hachassidus. Since the rebbe of the other chassidim is not doing that, it must be that they are talking about the Chabadtzker Rebbe.
This kind of thinking is much more common in religious groups than people realize. We assume everyone means the same ‘Gedolei Hador’ and ‘Those that brought Torah to America’ and ‘Talmidei Chaachamim’ and ‘Klei Kodesh’. But there is a lot of difference. The more obvious the different meanings, the more we tend to think that the other groupthink is out of the pale. But these labels really carry no weight. You can even refer to your local dropout as the Gadol Hador. It won’t have any effect.