Hello Mateusz, Tom,
I am relieved to see I am not the only one struggling with programming. With my huge knowledge of C

I have been checking the M119 supplied by Tom (thanks), and I do not understant on line 78 :
while (TotalCut < Bottom)
In my understanding, TotcalCut being 0 and Bottom a negative value most of the times the program will never enter the loop and then no tapping... (for the Jim version it seems correct to me) that would explain why it does not do anything on your machine or mine.
But just changing the it to while (TotalCut > Bottom) seems not enough, I tried it, machine had a very strange behaviour with Spindle going to very high RPMs (max of it actually 6000) and Z axis maybe trying to follow but I am not sure about that.
For what I see, the F parameter is not even used in the code, but S is, this is maybe why you need to activate the spindle before. Maybe replace Fxx by Sxxx will be enough in your case.
Cordially
Francois