NPN transistor as switch (actual voltage readings)
First, you are not in saturation. Simply having VB higher than Ve is not enough. Saturation requires supplying enough current to exceed the current needs of at the collector, minimizing the drop across VCE, and taking it out of the linear region. A lower value base resistor is needed.

Second, you need to account for leakage currents. It will never be zero, as nothing is perfect. 20 mV may seem large but its not really. But again, a stiffer pull down resistor would help minimize it.