The above applies to third mate’s situation on the Exxon Valdez
When there is a pilot aboard the pilot navigates by looking out the window and converting what he sees into a mental model which he then uses to navigate. This requires familiarity with the area and experience in making the mental conversion.
The third mate on the other hand can not do this. Instead he uses a much simpler method. Using the paper chart the mate takes measurements (bearing and ranges) and enters that data into the model (the paper chart is a simple model of a small part of real world) and gets an output - the ship’s position.
This is why the third made cannot “start turning back towards the lanes when abeam…”. The captain is using the language of the pilot’s method but the third mate is not a pilot. Instead, when abeam the third mate plots a fix on the chart. That’s the only way he can make sense of the situation.