Technically, the .38-200 was never British military nomenclature, it was the .380 Ball Mark I. The example you show (second from right) is a .38-200 as it is a commercial round. The first attached picture shows a Ball Mark I and a Dummy Drill D Mark I.
The .380 Ball Mark II (your right hand round) of course has a 178 grain bullet.
Second picture shows a series of Mark II rounds.
Ball Mark II (CN envelope)
Ball Mark II (GM envelope)
Drill D Mark II
Kynoch contract Drill/Inspection Mark II
Blank L Mark IT