I was always taught that MAYDAY means ..... I am in imminent danger and in need of immediate assistance.
whereas PAN, PAN, PAN means.... I have an urgent message to transmit.
There are no hard and fast rues as to when and where you make each call. The type of transmission rests entirely with the Aircraft commander / Captain.
In my view, they had a serious emergency here. They knew that a bird had gone down the intake, causing a serious fire. The engine fire was extinguised by securing the enging and then chucking a heap load of foam down it. They didn't know if there was any other damage to the aircraft following the bird strike. They didn't know if the fire had spread to other parts of the aircraft. They had a load of scared and frightened passengers on board.
They did exactly the correcr thing in maintaining the MAYDAY.