Learn English – “on (the) event day” and “on (the) celebration day”

As usual with definite vs. indefinite article, the definite article implies a specific day and the indefinite article does not. However, in context, even the indefinite article can mean "today".

So: If Tom is not at work today and Mary asks "where is Tom today?" either answer could be used and both would be understood. But if, at the end of the month, someone asks why Tom billed fewer hours, then "he took a day off" is correct and "he took the day off" is not. BUT, another correct answer could be "remember? June 15 was his son's birthday and he took the day off".

Whether to use "a" or "the" depends on what you are trying to say.
If you habitually rest on a particular day, then you would use

the day of rest

and refers to a particular day one does not work and can be understood to be naming that day as a "day of rest"

a day of rest

refers to a day, in general, when one does not work

"A day of rest" is not necessarily "the day of rest", but
"the day of rest" is definitely "a day of rest".

In your book analogy sentence sounds a bit awkward

I like to write stuff on the book when I annotate a book.

You are actually using the shorthand of

I like to write stuff on the book (that I am annotating) when I annotate a book.

and the meaning is

I like to write stuff on the (cover of the) book that I am annotating.

Using "on" implies you are writing on the outside of the book in your context. If you were writing regarding the contents of the book, you might use "about"

I like to write about the book (I am annotating) on its cover.