All pumps on pressure washers carries valves and seals which play a major role in maintaining output pressure. If the oil in the water pump looks milky, it means your seals are gone and water is getting into the pump (very bad), if you have minimal or no pressure it's your valves. Some units are servicible where you remove the bungs holding the valves and simply replace them with new ones (comet, cat, BEpressure etc. are like this) sadly many are not.
Surprisingly even the "commercial grade" washers need decent pressure from the pipe or they won't build pressure at the tip either (these are 3500-5000psi units). If the engine was running and there is no water flow to the pump, that would kill it alright. Trying to suck/syphon water from a barrel is a no no! Also leaving the engine running and not spraying for over 1 min causes the water temp in the pump to rise significantly. There is normally a release valve that lets the water out when this happens on most models but by that time the valves and seals have taken some damage. Hey at least you still have a working engine, make a go-cart or something and use that on it.