Why does the half moon appear to have a "nose"?
Here's an image I captured from a Moon globe app on my iPad.



From your description, I think you may be referring to the southern rim of Mare Tranquillitatis, the Sea of Tranquility, perhaps extending a little into the Rimae Hypatia. This is an area which is lower than the surrounding mountains and craters, into which the shadow is encroaching.

Incidentally, Mare Tranquillitatis is the region where Neil Armstrong and Buzz Aldrin landed on the Apollo 11 mission!