something strange,
looking at your pic the black fluid seems to go not vertical which is strange, i would expect the trace to be more vertical
i found this pic to explain it a bit,
you see the push rod, well the tolerance in diameter between the hole in the cylhead and the rod gives you the lubrication, using the same engine oil that fills the cylhead
point is that the oil accumulated will be very little and will not go anywhere
to have that oil spilling from the mech rod there might be three reasons:
1- there is too much oil in the chamber and it exerts so much pressure that it has to leak, and provided there is not a seal in that rod, it comes out (here it comes my recommendation to wipe clean it and see how much it leaks)
2- there is a strong deterioration of the balance mech that actuates the pump membrane and it has eaten the pump wall so the oil has found a way out (i had a completely destroyed internal brass bush in my old pump, forces are small but very repetitive there)
3- your membrane is broken so some fuel comes into the chamber, mixes with oil and, being more thin, makes the fluid come out through the rod
the third one is the most critical, as the leak in the membrane will grow bigger every cycle, at the end you will have gasoline coming into the cylhead
question, can you tell that the black fluid smells to gasoline ?