Can capitalization be considered as a marker of definitness in proper nouns

Solution 1:

Does that mean that the capital letter is considered a marker for definiteness?

No. Often they are but you are trying to create a “rule” and there are no “rules” in English – only guidance. Some guidance may look like a rule, but it is not. The following is guidance:

Capital letters are used for

Proper nouns Beijing; Taj Mahal; John; Monopoly (the game).

The purpose of expressing respect “I spoke with Her Majesty”; “I spoke to God, and He spoke to me.”

For nouns and important words in media titles: Love’s Labour Lost; “The Man with the Golden Gun” “A Tale of Two Cities”; “Lost Boy found in Forest!”

For titles of rank: “The man in charge is General Smith.” (but “Many of the generals were asleep.”)

For titles when used for direct address: “I don’t know the answer, ask Dad.” (but “I’ll tell my dad what you have done!”)

At the start of sentences and lines of poetry.

In philosophy to distinguish between an item and the essence of that item: See Wikipedia Capitonym https://en.wikipedia.org/wiki/Capitonym

Some other cases