I've used one of those scrapers that takes razor blades to loosen the bits of moulding around the glass, kind of jamming the blade into the gaps. Then a tack lifter and a chisel to help ease the bits off. One you've got the mouldings off one side the glass should come out fairly easily.
It's an internal door, so does have the quadrant holding it in - one pane has a crack, but the other is fine. I am guessing the glass was put in anytime between the 50's - 70's (original door would have been standard Edwardian internal one leading into the kitchen from the hall). Apart from being horrible cheap 'frosted' glass, it's also definitely a safety hazard so I want to re-glaze with toughened and need to bring the originals down to the glaziers for them to measure up for new.
once you have lifted the quadrant a fraction with a narrow sharp implement, the tack lifter will wiggle it out and raise the fixing pins. Pincers will also help with the pins.
Ask the glazier about laminated glass. It does not have to be ordered, made to your size like toughened, it can be cut to size in the shop. But like a car windscreen (not like a car side window) it will craze but not shatter or break into small pieces.
It is also burglar resistant, but that won't be relevant for an internal door.