That's right. But since the cathodyne has a gain of less than 1 many of us do consider the other triode as part of the PI.
More on this:
The balance of factors in a long-tail inverter results in the gain from the input of that inverter to either grid being ~1/2 the amount you'd typically get from the same tube in a normal gain stage.
As Sluckey said, the split load's large cathode resistor creates feedback and a gain of less-than-1 (so a slight signal loss from grid to either output). Add the other triode in the envelope in front, and you get 2 output signals with gain from grid to output ~1/2 the that of a normal gain stage,
because the 2 load resistors are half the value of a typical gain stage. Essentially, the split-load takes the plate load, slices it in half and slides one half down to the cathode.
Net result is the gains of the 2 circuits will be similar, as long as you use that pre-gain stage with the split-load inverter. You'll also notice in most old amps using the split-load that the feedback from the speaker (normally injected right at a long-tail grid) is returned to the pre-gain stage in front of the split-load inverter.
Bottom-line, you'll be using 2 triodes for your phase inverter or you'll need to boost your preamp signal driving the split-load by adding ... wait for it ... another gain stage.