What worked for me was, placing the head of a closed pair of pliers behind one of the three pins(that are used to turn the fitting around) and lightly tapping the back of the pliers' handles with a hammer to put more pressure/force on the pins than any of our hands could, and thereby release the jamming/locking inside the fitting.
The two pins, as you know, belong to the lamp. So pull them out. The (“MR16”) lamp socket will probably be damaged do you’d need a new lamp socket.
HOWEVER - the halogen lamp that you are trying to replace are obsolete. You can’t buy them any more.
You need to move to LED lamps. Unfortunately you will also need to change the existing 12volt transformer (up in the ceiling ) for a LED driver. Or remove the transformer altogether and move to a 230v GU10 lamp. You’d need to change the bulb holder if you go that way.
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