Morgan is a road car that can be used on the track. Caterham is a track car that can be used on the road.
I tried a couple of Caterham cars before buying the Roadster in 2012. Your perception matches mine perfectly. The Caterham is for someone who wants to do a lot of track days and other forms of off road events. The Morgan is capable on a track but it is better at getting to the circuit than on it.