Just a minor correction and it is sometimes difficult to find the real truth as many sources are incomplete and there is a curious use of terminology that sometimes confuses the issue.
The illustration used is not a reciprocating piston meter but what is referred to in Europe and many other areas today as a “Rotary Piston” meter.
It was first introduced in the 1860s by Tylors of London as the “British Patent Rotary Water Meter” and only suitable then for the rotary (rotary vane meter) market (direct plumbing) i.e. the market normally served by inferential meters, notably at that time the single jet meter (probably the first water meter).
The reciprocating piston meter was patented by Thomas Kennedy in 1824 as a water meter for indirect plumbing. This is a Positive displacement meter. Though casually referred to as positive displacement the rotary/oscillating piston meter is only semi-positive as open flow paths exist through parts of the measurement cycle.
In the early 1900s the three classes of meters were inferential or rotary vane meters, displacement meters (rotary piston/nutating disc) and positive displacement meters which included many different technologies but the benchmark was then the reciprocating piston meter.
Over time, and most probably in the 1945-1960 period in the UK and other markets for this meter, it became referred to as a “Rotary Piston Meter”. This was most probably at the time when the positive displacement meters were eliminated from water metering mainly due to complexity size, pressure drop and cost.
In America the term rotary piston meter was used to describe the Crown meter (example in the Smithsonian) invented by Lewis Nash. This is visually more like a gear meter but the term really suggests that this meter was one suitable for both direct and indirect plumbing and not, as some sources suggest, to discriminate it from the reciprocating piston meter. Hence, to avoid confusion with the Crown meter, in the US this meter is referred to as an oscillating piston meter.