What is meant by 'local' in local oscillator?
You got that wrong:
The "local" in oscillator doesn't describe the kind of oscillator used. It could just be an LC-tank, it could be a crystal-derived oscillator, it could be something synthesized from a reference clock or something recovered from the data stream received:

The "local" in oscillator refers to the fact that it's what the mixer uses locally to mix down or up, as opposed to the oscillator at the other end of the communication, which simply isn't the same oscillator.