NavList:

Being a sponsored scientific expedition, Cook and his crew made conscious efforts to be precise about their measurements. Most (though not all) longitude & latitude sightings were described in relation (or lack thereof) to land. It's for instance clear that the coordinate readings in the above excerpt were not made on land, but in the middle of sea. Others, however, were recorded and refined on land:

Monday 12/11 (1769): Early in the morning, we stood in with the land, seven leagues to the westward of Doubless Bay, the bottom of which is not far distant from the bottom of another large bay, which the shore ... being separated only by a low neck of land, which juts out into a peninsula that I have called Knuckle Point. About the middle of this bay, which we called Sandy Bay, is a high mountain, standing upon a distant shore, to which I gave the name of Mount Camel. The latitude here is 34 degrees 51' S and longitude 186 degrees 5'.

I also read in a secondary source that the majority of lunar readings were made by Charles Green, assistant to Astronomer Royal Nevil Maskelyne (the lunar guy himself). To be precise, he recorded hundreds of sets of lunar observations for locations to be incorporated into new maps. This would not have been possible in mid-ocean ship conditions (I think?).

In other words, most of the time, Cook was clear about location: observations at sea meant observations at sea, those referring to capes or beaches or islands are usually accompanied by descriptions of local flora & fauna, meaning that the ship made a stop on land. Cook is also very good at being specific about the layout of the land itself: islands often have western and eastern extremities, as does beaches and even inland sea routes (e.g. entrance to Queen Charlotte's Sound). And of course, much of this coordinates were meant to be incorporated into new maps, and so maps made immediately after the expedition are good sources for location. 

Nonetheless, the question about precision still remains: exactly how precisely can we assign modern/accurate coordinates to Cook's coordinates, so that the error calculation actually mean something. Admittedly, this problem made me break out in cold sweat throughout the project--especially when dealing with measurements based on dastardly vague "cape-this" and "cape-that". Here I assume that "cape", "bay", "point" refer to shores where Cook's ships harbored and where observations were made. I threw out locations when such "cape" "bay" or "point" referred to a wide stretch of land. For the locations I worked with,  "north" vs "south" of a "bay" or "beach" is significant only in seconds of longitude. However, because Cook's measures were almost always precise only to minutes of longitude, I decided to allow this imprecision in seconds of longitude. This means I had to round all error calculations to minutes of longitude rather than seconds (unfortunately, not sure if this is acceptable methodology).

The problematic nature of historical data demands careful handling. My experience with Cook's journals is definitely a learning process, and frustratingly layered by multiple attempts to redo things. To be honest, the "final turned-in" product is still not good, and the data could benefit from additional cleaning and scrutiny. But I do believe that there is enough resources and primary documents out there to render this sort of error analysis meaningful. It just requires a lot of research and care.