Given a triangle I draw circles around each vertex. I chose the radii of these circles so that they are all mutually tangent. There is only one way to do this. I extend these tangents. They concur at a point. Is there a name for that point?
Looking at the Encyclopedia of Triangle Centers I didn't find a match, though this has the right theme. So the 1st Ajima-Malfatti point of the outer triangle is the point-to-be-named of the inner triangle (EDIT: on closer look, I'm not sure that it is exactly). And as it happens my motivation does come from circle packing from a triangulated graph.
Incenter. Your construction is an alternative description of the center of the inscribed circle.