definite article - Use of "the" before the first part of Of-phrases

There is a similar question with a similar title, however it then focused on two specific examples. The main question is when we should use "the" before an Of-phrase.

Is "the" used for the first part or the whole phrase? for example, is it [the history ] of the internet. Or the [history of the internet ].

Some examples by searching google:

• the International Day of the Girl Child


• the study of Pattern Recognition


• The Charter of the United Nations


• 
  

  More than 60 United Nations Member States have never been Members of the Security Council.

  
  



• 
  

  the Word of the Day

  
  


• The history of the Internet


• Transactions of the American Mathematical Society


• Proceedings of the Royal Society


• Types of problems and tasks

For example in

Machine learning is a subfield of computer science that evolved from the study of pattern recognition and computational learning theory in artificial intelligence.

Why "the study". Please note there is not much context for the "study" and I suspect the possessive construction has a role.

Apart from these examples, I feel in Of-phrases the domain of objects become limited, and it allows to use "the" for the noun before "of". For example you can't say "the mouse"! which mouse? but you can say "the mouse of a computer" because in the domain of a single computer, mouse is specific.

For another example we may say "The manager of a shop" (even without any context) but not "the customer of the shop" (again without any context). maybe "the customers of the shop" or maybe just "customers of the shop". "the door of the house" but not "the window of the house", because without any context we don't have one window but one door. Similarly, "the head of a human" but not "the hand of a human" (maybe the left hand of a human).

Does this analysis creditable?