Max, it looks to me as if both the automatic and manual oil pumps are actually air pumps that pressurise the tank, allowing oil to flow through the nozzle onto the chain. Otherwise it would not have cured both problems when you made the tank air-tight so it could be pressurised. The tank cap does not seem to be vented, which supports that theory of what is going on. The downside would be that after you switched off the saw, oil would keep oozing out onto the chain for a while. There may be a small air leakage path to speed up the loss of tank pressure after shutdown.

If all that is wrong, would some Echo chainsaw afficionado put us right please?