Technically, if the outlet isn't at ATM then the tube should be reversed to put the metering valve on the outlet side, not inlet. Like if using in line or diffusion disc. Suppose it's fine for reactors. Not exactly sure why, but that's the recommendation as I understand it.
Never mind, seems they changed that a bit.....
float control valve at the outlet side is for flow rate monitoring under specific pressure setting(pressure in tube other than atmosphere pressure).
float control valve at the inlet side is for flow rate monitoring under normal atmosphere pressure setting(pressure in tube is 0 psig, gauge pressure, or, 14.7 psia, atmosphere pressure)
and the porter/parker flow meter in the picture is capable to reverse the inlet/outlet by turning around the glass tube.
if use this flowmeter with reactor, the valve should be at inlet side because the pressure to push co2 into reactor is close to atmosphere pressure, and if use with higher back pressure type diffuser, the valve on flowmeter should be on the outlet side because the pressure inside the tube can be adjust to a specific number according to the flowmeter manufacture.
when you mention matheson rotameter I think you are on the right direction but the details can be found in this page, for flowmeter FM1050, you can check flow rate(air) of different model of glass tubes, under different pressure setting, different floating ball material. BTW, the tube models useful to us are 610A, 600, 601.