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"六日" means "the 6th day of the month".  
"六日間" means a "time period of six days".

In a textbook, I found this example: "六日目の朝". How would you say that in
English?

I _think_ that "六日目" means an abstract kind of "6th day"? But, what is that
"6th" day relative to? What are some examples that use "~日目"?
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counters
Usage of "六日目", "七日目", "八日目", etc.?
It's just what you probably already guessed:

> 六日目の朝  
>  the morning of the sixth day

_relative to the beginning of the story/narrative_ (i.e. it doesn't have to
mean "6th of the month"). You know how to use `th day`:

> The **first day** we went to the park. [Blah blah blah.] On the morning of
> the six **th day** [we found ourselves tired from all the park-going and
> decided to start smoking weed on an empty stomach].

As far as I can tell it works exactly the same in Japanese.
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You are correct that 六日目 means "the 6th day". Thus, 六日目の朝 is the morning of
the 6th day. Relative to what? Relative to whatever the context where it
appears.

>   * 旅行の六日目に海に行った。 → On the 6th day of our trip, we went to the beach.
>   * 裁判の十日目 → The tenth day of the trial
>   * クルーズの二日目 → The second day of the cruise
>
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As you stated, if you used 六日 alone it refers to the "sixth of the month".
However, you may want to refer to the sixth day of something, where something
will be either defined by the context or explicitly expressed.

In that sense, I think that ～日目 is almost the same that 第～日, the latter being
more formal to my opinion.

I did not have another example in mind, but, for example, wikipedia sums up
the beginning of the Book of Genisis the following way:

>   * 1日目 暗闇がある中、神は光を作り、昼と夜が出来た。
>   * 2日目 神は空（天）をつくった。
>   * 3日目 神は大地を作り、海が生まれ、地に植物をはえさせた。
>   * 4日目 神は太陽と月と星をつくった。
>   * 5日目 神は魚と鳥をつくった。
>   * 6日目 神は獣と家畜をつくり、神に似せた人をつくった。
>   * 7日目 神は休んだ。
>

From
[天地創造](https://ja.wikipedia.org/wiki/%E5%A4%A9%E5%9C%B0%E5%89%B5%E9%80%A0)

If I would have wanted to state clearly that it is the sixth day of the
creation of the universe, I could have said, 天地創造の6日目.

In [one](http://bible.salterrae.net/kougo/html/genesis.html) of the
translations of the Bible, you can see that 第～日 is used.
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