I still do not get the concept of a state Managed crypto-currency. How is it a crypto-currency if it is created by and run by a state?

Just because it is an electronic currency, that does not make it a crypto-currency. The U.S. Dollar is both a physical and electronic currency (I can transfer dollars and make payments in dollars electronically). But it is definitely not a crypto-currency because it is sponsored by the government of the United States and managed by the US Federal Reserve.