What is happening when digitalWrite(pinname, LOW) is invoked after pinMode(pinname, INPUT_PULLUP)?
It's pretty explicitly stated there in your quote: if you write a LOW when in INPUT mode it disables the pullup resistor. That's all it does.
pinMode(4, INPUT);
digitalWrite(4, HIGH);

is exactly the same as:
pinMode(4, INPUT_PULLUP);

And
pinMode(4, INPUT_PULLUP);
digitalWrite(4, LOW);

is exactly the same as
pinMode(4, INPUT);

Of course this exploits a peculiarity of the AVR's IO block, but any decent core will emulate that functionality for compatibility with older code.