In the photo, the left side has a red zig zaggy line. That is showing that the two switches are connected in a way. So that is why we need only a SPDT switch. Only three pins. Not four pins. The yellow boxes show where the switch solder plates are.
The two white wires are connected to the 'ON' switch. So when the switch is being pushed, it is substituting the original 'ON' switch.
The white wire on the left and the blue wire on the bottom are for the 'OFF' switch. When the switch is released or out, the new switch is substituting for the 'OFF' switch.