You definitely have to define bitcoin as money. It has all the required features, and it is used in exchanges.
That's not a good reason to classify bitcoin as money. Bitcoin has additional features that money lacks. We don't have a word for it yet but what ever Bitcoin is, money is a subset of that.We have words.
Compared to paper fiat:
Possible to transfer over the net.
Can be backed up.
You need a computer
What kind of properties does bitcoin have that makes it not money?