I notice that you did not understand the referred answer, so I will try to explain it with other words.
The Groundspeak API is what Project-GC uses to retrieve data from Geocaching.com. Your interface is www.geocaching.com, ours is a text based protocol.
When you add data into the "correct these coordinates" which is what pinkunicorn refers to as "Corrected coordinates field", you are adding it into a field that we can not so easily retrieve via the API. Therefore, Project-GC will (normally) not know about it.
But if you add the coordinates into your Personal Cache Note field, Project-GC can retrieve the information, and then also handle the information.
To explain why we can use cache notes and the corrected-field then.
For cache notes, we can ask the api "Get all my (Stjärnvik's) cache notes". Simple as that, now we just need to parse the information.
For corrected coordinates we have to ask the servers, do I (Stjärnvik) have coorrected for this cache? It would mean, that we reguarly (daily), would need to ask this for every Unknown/Multi cache in the world. I don't know the number, but I assume around 500.000 requests per user per day. As you probably understand, this is not doable.