Jens_P
Since the problem disappeared after engine shutdown, this appears to be a loose object in the raw water circuit. Could even be a piece of plastic that has covered the water intake in the stern drive that has fallen off while engine was stopped.
Faulty rubber impeller might be a problem. The pump is found inside the sterndrive. Checking or replacing means that you have to split the sterndrive to gain access. This is not the first component I would have checked.
A well known problem is the heat exchanger getting clogged as shown in your picture. This problem builds up slowly and will not disappear just by stopping and starting the engine.
