How a country is categorized as developed or developing?
The terms developed and developing are UNDP terms.

Read the paper at the link below - it goes through the UNDP, World bank and IMF classifications and it's an IMF working paper so i'd class that as referenceable. At the very least it will point you to better sources.

https://www.imf.org/external/pubs/ft/wp/2011/wp1131.pdf