Is there a name for the point where a Peltier cooler loses power and the cold side becomes warmer than the temperature around it and vice versa after power has been lost?
It's called poor thermal design. The Peltier devices reach a maximum Q or heat energy transfer when the cold side goes into an infinite heatsink.
Without a good heatsink, the cold and hot side reach an equilibrium hotter than the ambient.
Therefore consider your thermal conductance on both hot and cold side to be of utmost importance. This requires a good thermal heat spreader with grease and forced air 1 to 3 m/s velocity over the heat spreader fins. e.g. a CPU cooler for the hot side and larger mass with low Rth on the cool side.