For a hopefully simpler explanation, refer to the transformer wiring diagram in the first post.
In particular the
Secondary Volts table connections of X1, X2, X3, X4.
Now remember that a transformer doesn't care which windings you use as a primary, secondary, tertiary, or whatever. The relative voltages are determined by the turns ratios between the windings.
1) For 120V on the secondary, the two windings are wired in parallel. This implies that the windings are identical; if they were NOT identical there would much current from one winding flowing thru the other one. Not very desirable.
2) Now for 240V, the windings connect in series; two identical 120V windings in series for 240V. If this is an output it would be considered 120/240V two-phase.
3) Since a transformer doesn't care where the power comes in or goes out, you can apply 120V INPUT from the X2/X3 connection (the center tap) to either end of the winding, X1 or X4.
4) The 'other secondary' (other half) winding is still connected in series with the winding where you supplied power.
5) Since they are in series (with the correct phasing), you get 240V across the winding ends (X1 to X4).
As
@DaveE pointed out, this configuration is called an 'autotransformer', or rarely an 'autoformer'. There are commercial products used as test equipment that can supply a continuously variable voltage from zero to above line voltage. This is done with a sliding contact that can be positioned at each turn of the winding. Two common brands in the U.S. are 'Variac' and 'Staco'. Try an internet search for them for more info.
As for the derating, I chose 3 because I was (and am) too lazy to look it up/investigate in detail. It is based on the fact that the wire used for the powered part of the winding (and the Iron core) must support the total power withdrawn, plus losses. Perhaps the derating is closer to 2, but I chose to be conservative.
This ended up a longer post than I expected! Hope it helps.
Cheers,
Tom