Yeah, you don't have it completely right.
Comment by Bean is pretty on the money, but here's the thing.
First, clawing has to do with what happens to the hand when the patient is asked to straighten/extend his fingers.
If the ulnar nerve is damaged then you get ulnar claw (2nd/3rd digits straight, 4th/5th digits extended at the MCP and flexed at the PIP and DIP). This is due to loss of the ulnar innervated lumbricals that leave the radial extensors unnopposed.
If the median nerve is damaged, then you get the same deformity - but only with the 2nd & 3rd digits instead. This is median claw.
If the patient is asked to make a fist, and said patient has median nerve damage then you get the hand of benediciton (2nd/3rd digits slightly arched, 4th/5th digits completely flexed). This is due to loss of the FOREARM flexors innervated be the median nerve.
The confusion stems from the fact that the Hand of Benediction/ Pope's Blessing looks very similar to Ulnar Claw... The similarity is only superficial though, the mechanism resulting in the deformity is different.
So Pope's blessing refers to a PROXIMAL median nerve injury and is seen upon MAKING A FIST.
Claw hand, without further descriptive modifiers, refers to a PROXIMAL unlar nerve injury and is seen upon MAKING A FIST.
An ULNAR CLAW is DISTAL ulnar nerve damage demonstrated upon finger EXTENSIOM.
Likewise, a MEDIAN CLAW is DISTAL medial nerve damage demonstrated upon finger EXTENSION.
So there's 4 diffent hands, one for proximal and distal nerve injury of both ulnar and median nerve. 3 have the word claw in there name and 2 of the hands resemble each for dissimilar reasons... hence, why it's so confusing and generally poorly understood.