To be clear, they're not named differently according to where they originate. They are named differently in different countries. The same storm system that will be named a tropical cyclone in Australia will be named a typhoon in Asia.
Cyclone Debbie is what we in Australia are calling it. It's the most important name because it's hitting us, and not another country. If it were hitting another country, I would care about what that country named it, but we in Australia might still refer to it as a cyclone, because that's what we call them.
To be clearer still;
This may be true of common parlance used by locals talking in a bar or on a street corner.
And I would be unsurprised (although a bit disappointed perhaps) to hear locals around here call your storm 'Hurricane Debbie'. But if they had learned about it by watching the news and weather on TV they would have been told it was called 'Cyclone Debbie'.
But as far as professionals are concerned ...
The National Ocean Service (a branch of the National Oceanic and Atmospheric Administration) has
this to say about that.
The only difference between a hurricane, a cyclone, and a typhoon is the location where the storm occurs.
I want to note that the emphasis is
theirs, not mine.
None of the meteorologists living and working here in the U.S. are going to call Debbie a hurricane, even though they are speaking here in the U.S.
That's because the convention in meteorology is to call a storm of that sort which occurs in that part of the world a "Cyclone".
Another evidence of the formality of this distinction can be seen in the fact that the lists of designated tropical cyclone storm names are separated by geographic region as well, with care taken to ensure that duplicate names do not exist across those regions. This is all sorted out and made official by the World Meteorological Organization.
Hence, although there can be a tropical cyclone storm named "
Hurricane Debb
y" in the Atlantic, there can never be a "
Hurricane Debb
ie", because the storm name "Debb
ie" is reserved for "a tropical cyclone storm
below 10°S between 90°E and 160°E".
Due to this formal assignation
any tropical cyclone storm named Debbie would always be
officially named "
Cyclone Debbie".