The article we're discussing explains that Japan has the best passenger rail system in the world, and which happens to be privatized, along with privately owned track. So which one is it? Go figure.
Then again, given the UK experience of going all-in on the "vertical separation" and privatization path, perhaps one shouldn't blame them.
Technically, yes, the JR's are private companies.
But track construction is generally done by a government construction company financed with Japanese sovereign debt. The completed tracks are then long-term leased to the JR's at favorable rates.
Is it really a private company if the key capital outlay is done by the government and given to you with a sweetheart deal? ehhhhhh.... you can call the operator company private, but you're being dishonest if you call the system privatized.