Using an 8 ohm speaker in on a 4 ohm tap will result in a lower volume. Using the power formula, P=V2/R, the apparent power will be half. ...
This is logical, but it assumes "all else unchanged." Unfortunately, all else
does not remain unchanged.
If the supply voltage and operating point allow, the plate voltage swing ("V" in the formula) gets bigger with a higher load impedance.
If the output tube could manage the same plate current swings, then with a larger load impedance Ohm's Law shows the plate voltage swings will get bigger (same-I * bigger-R = bigger-V).
So in your power formula, R got larger (power goes down) but V also got larger (power goes up). But the same-current assumption above may not hold...
We can't know the net effect without more information. What we can know is that if the original load impedance was the one impedance for the tube, supply voltage and class of operation which delivers maximum output power, then any deviation above/below will result in less clean output power.
... Using the power formula, P=V2/R, the apparent power will be half. Look at the data sheets from hammond, you will see that the 4 ohm tap has the sqrt of 2 lower voltage than an 8 ohm tap, and half the voltage of a 16 ohm tap.
This also seems to be logical proof an 8Ω speaker on a 4 Ω tap will give half-power (if you do the math for the indicated secondary voltages, 8Ω on the 4Ω tap does give half-power compared to 8Ω on the 8Ω tap).
The flaw here is that the secondary voltages assume the same voltage swings occur on the primary regardless (the secondary voltages are equal to the primary voltages transformed by the step-down ratio from primary to secondary).
However, the changed secondary impedance (8Ω on the 4Ω tap) result in a change of reflected primary impedance, which then changes how the output section performs. Again, we can't conclude any rigid relationships without more info on the output section supply voltage, current capability and class of operation.
The approach used for the wrong assumptions above treated the output section somewhat like a transistor output stage... That is, it assumed the output section could throw any current required to maintain the output voltages indicated at the secondary of Hammond's diagrams.
By using that key assumption, you would also conclude that attaching a 4Ω load to the 8Ω tap gives double-power. And those kind of games get played with transistor power amps (attaching lower loads to get louder/more-power output), right up to the point the output transistors melt down.
Tubes don't work that way, so going lower or higher from optimum will result in less output power. This is graphed on the more complete data sheets, even though the specific numbers (loads, power output, distortion %) really only apply for the single operating conditions listed in the graph. However, the general principle always applies.