What's the difference between ぬきで & ぬきの and 入りの & 入りで?


I can't find the difference and in the book that I use there is no
explanation.

* * *

Context: It is about ordering food. I have two sentences:

> ミルクと砂糖 **入りの** 紅茶はいくらですか。  
>  ホットコーヒーをミルクと砂糖 **入りで** ください。




Generally speaking, ～で is always adverbial (i.e., modifies a _verb_ ), whereas
～の always connects to the following _noun_. In English this distinction is not
always clear because "with ～", "to ～", "in ～" and so on [can modify both a
noun and a verb](https://japanese.stackexchange.com/a/17739/5010).

  * ワサビ抜きで寿司を食べた。 I ate sushi without wasabi; I ate, without wasabi, sushi.
  * ワサビ抜きで寿司 (sounds like an incomplete sentence fragment because there is no verb for 抜きで to modify)
  * ワサビ抜きの寿司を食べた。 I ate sushi without wasabi; I ate wasabi-free sushi.
  * ワサビ抜きの寿司 wasabi-free sushi (noun phrase)

When you order something, you can say, for example:

> ミルク入りでください。  
>  ミルク入りでお願いします。  
>  With milk, please.

...because ください/お願いします is a verb. You can omit this verb and say this simply:

> ミルク入り **で** 。  
>  With milk (, please).




Generally speaking, ～で is always adverbial (i.e., modifies a _verb_ ), whereas
～の always connects to the following _noun_. In English this distinction is not
always clear because "with ～", "to ～", "in ～" and so on [can modify both a
noun and a verb](https://japanese.stackexchange.com/a/17739/5010).

  * ワサビ抜きで寿司を食べた。 I ate sushi without wasabi; I ate, without wasabi, sushi.
  * ワサビ抜きで寿司 (sounds like an incomplete sentence fragment because there is no verb for 抜きで to modify)
  * ワサビ抜きの寿司を食べた。 I ate sushi without wasabi; I ate wasabi-free sushi.
  * ワサビ抜きの寿司 wasabi-free sushi (noun phrase)

When you order something, you can say, for example:

> ミルク入りでください。  
>  ミルク入りでお願いします。  
>  With milk, please.

...because ください/お願いします is a verb. You can omit this verb and say this simply:

> ミルク入り **で** 。  
>  With milk (, please).

