Well, that's unexpected. The diode can only reduce the power by 50% in any full cycle. The only way it could work as specified is if they used a PTC heating element, which could draw 120W when cold, but its resistance goes up enough when hot for it to drop to 20W at 50% duty cycle. In that case, when you reduce power, the element cools and it draws more current till it reaches equilibrium, only slightly cooler.
To be certain what's going on,you'll need to measure the current it takes, preferably using an isolated current probe on a scope. Alternatively, first measure the element resistance when cold (across mains plug L and N pins with the button held down), run the iron till its at max temperature, unplug it and immediately start measuring its element resistance (across mains plug L and N pins with the button held down), repeating the measurement every ten seconds as it cools till the solder on the bit solidifies.
The diode may be causing issues for your control circuit - You could try putting a 40W incandescent bulb in parallel with the iron so the TRIAC's load isn't Hi-Z on one half cycle, to see if that lets you get a usable control range.