No, you did not misunderstand. D3 is a PWM pin, D4 is not.
For D4, analogWrite(4, 120) shows a constant low signal on the pin 4, analogWrite(4, 130) shows a constant high signal. This indicates that D4 is not a PWM pin.
The problem is in the schematic which marks D4 as a PWM pin (D4*) and the silkscreen (print on the board) is based on that. https://www.arduino.cc/en/uploads/Main/arduino-micro-schematic.pdf
And I have this one on my PC
Arduino-Micro_Rev1j_sch.pdf (91.9 KB)
PS
If you look at the schematic, only pins (inside the processor) that contain OCxy are PWM capable pins. E.g. for pin 3, it has OC0B).