[G8HQP Dave]

As I keep saying, the lower device has voltage gain if you use a big enough anode resistor. Just to pick some numbers at random, assume gm of 5mA/V and mu of 20. An anode resistor of 20k means a voltage gain of around 100. 20 of that comes from the upper valve, and 5 of that comes from the lower valve, approximately. 10mV in would become 50mV at the 'join' and then 1V at the output. What would you call an amplifying stage with 10mV in and 50mV out (or 50mV in and 1V out)? I would call it a voltage amplifier.

I have never said that the current is amplified by the upper device. I, and others, have said repeatedly that the voltage is amplified by the upper device.

You seem fixed on the idea that a triode is a transconductance device. It is also a voltage amplifier. You need to use the most appropriate model, not just stick to just one all the time. Your simplified model of the cascode will not enable you to derive the full equations you quoted; to do that you need to take account of everything which is going on.