The examples you mentioned come pretty close. Bitcoin is a “virtual currency”, just like airline miles and REKA money. The main difference is that it is generated by an interest-group rather than a company or organization. Bitcoin is created as payment for people who provide the internet servers to power the network. The more computing power you make available, the more Bitcoin you earn. In that way, Bitcoin is much like a community currency because, at least initially, you have to provide a service in order to get it.
Because Bitcoin is only generated or “mined” as more computing power becomes available, there are physical limits on the amount of Bitcoin which can be generated (unlike airline miles, which are created on demand). As with community currencies, the only real value of Bitcoin is in people’s willingness to accept it in exchange for genuine currency or services. There is no economy backing this virtual currency.
The hype around Bitcoin lies in two factors: Firstly, just like airline miles or hotel loyalty points, Bitcoin can (currently) be transferred across borders without limitations and without the need for financial services providers, making it a cheap and practical medium for transferring wealth across borders.
Secondly, the “blockchain” technology which powers Bitcoin records every transaction which has been made with each specific Bitcoin. This is roughly the equivalent of being able to read every fingerprint on a banknote dating back to its creation. Although Bitcoin can currently be mined and spent anonymously, the technology it is built on could eventually be used to impose full control over financial transactions.