Quality wise you will be very happy with your watch forever....even when you see/understand the quality. L&S is truly one of the best made watches money can buy. Wish you many happy moments with your watch !
...or then the screw inside the pusher is getting loose (pusher turns) and then one day you will see no pusher ! It is quite common problem. Qualified watchmaker would be able to find the right answer to your question. Have a nice day.
I wouldn't be able to name it ...but it is a tool to separate the balancestaff from the "plate"...or some people use it also to separate the cannonpinion from its axle...!???