I have searched but couldnt find anything on this topic specifically.

Lately I have been starting the engine when it is on 2 bars- in the middle between warmed up and cold. It happens just because of timing, I park my car and then have to leave later when the car hasn't completely cooled down.

I have heard that this can cause damage to the engine? Why is it causing that and what can it potentally cause?

This is kind of the same situation as not warming your car up all the way before shutting it off again.

 

I don't know of any damage that will happen. My Insight turns itself off whenever I come to a stop and then starts again when I go to leave the stop. I don't think Honda would design it to do that if it would cause damage to the engine. I am guessing you are o.k.

 

Was it from some "old wife"?

It's not the engine that "might" get harmed. It's the exhaust system that can accumulate condensed moisture from the exhaust gases. If the system doesn't get heated up to operating temps, the water can collect and if left too long, this can begin the corrosion process. You would have to drive the car day after day down to the corner store 2 blocks away and back again and not drive it again till the next day, that is the hardest on the exhaust.
Sure condensation will also collect in the engine, but it's covered with oil.