Is a cyclone a tornado too?

No. A tornado is a small scale but violent vortex that forms as a result of a thunderstorm and are dependent on a parent storm.
A cyclone is a large-scale low pressure system orders of magnitude larger than a tornado. Cyclones can be violent but do not have to be.
Cyclones and tornadoes both have low pressure and cyclonic rotation, but they are classified separately as they operate on different scales and are powered by different mechanisms