How do you describe then the Joule-Thompson experiment? I.e. the expansion of a gas from (internal) pressure p1 to pressure p2 trough a throttle?
According to e.g. the book of Max Planck "Thermodynamik" (Walter de Gruyter, Berlin, 1930) (the argument can be found in numerous more modern treatments on thermodynamics, however, Planck certainly understood what he was writing about as he did his thesis on thermodynamics and got a formulation of the second law named after him.) one choses the enthalpies before and after passing through the throttle as H1=U1+p1V1 and H2=U2+p2v2. The process is called "isenthalpic" as H1=H2.
In my opinion, at best one of the two pressures can coincide with the external pressure, which may however even be distinct of the two pressures in the experiment.
PS: In my previous post I didn't assume any piston but only that of gas enclosed under pressure in a steel cylinder as is used for storage of gas.