Note first that it's not technically steam that forms, it's just mist, i.e., a lot of tiny drops of water drifting in air. read more
In the latter example, hot water ~120F would have 5x the molecules of water above the surface than the cold water at ~70 F. A lot of vapor would condensed above the already saturated surface of the cold water when the hot water is mixed in. The condensing mist would stop when the water and air are again at the same temperature (equilibrium). read more
Vapour and Steam are both used to refer to the gaseous form of water (there is a slight difference as per /u/browb3aten above - vapour should only be used below the critical point - above that it is a gas, or steam in the case of water). read more