Why don't we use ～なくて instead of ～ないで?

Question

Background

In my understanding that might be wrong, I think the ～ない form of a verb becomes an i-adjective. For example,

納{なっ}豆{とう}を食{た}べない人{ひと}が多{おお}いです。

where 納{なっ}豆{とう}を食{た}べない modifies 人{ひと}.

Furthermore, I think I should be able to make use of the following pattern as well,

目{め}が大{おお}きくて納{なっ}豆{とう}を食{た}べない人{ひと}が多{おお}いです。

or in the reversed order,

納{なっ}豆{とう}を食{た}べなくて目{め}が大{おお}きい人{ひと}が多{おお}いです。

Question

Today I learnt a new grammar.

砂{さ}糖{とう}を入{い}れてコーヒーを飲みます。

What makes me confused is the negative version as follows,

砂{さ}糖{とう}を入{い}れないでコーヒーを飲みます。

Why don't we use ～なくて instead of ～ないで as follows?

砂{さ}糖{とう}を入{い}れなくてコーヒーを飲みます。

Answer 1

ないで comes from the combination of ない negative verbal form and で which stands for way, means of doing sth e.t.c. ないで generally can be used in the same context with なくて but なくて is a plain negative form with the connector て.

The example you present is better off with ないで because it translates word-to-word to

" I drink the coffee, by the mean of not adding sugar", so "I drink the coffee without adding sugar".

The なくて version just connects things you do or don't.

"I do not add sugar and I drink coffee".

This could be easier understood with the following example:

私は学校に行かなくて山田先生に行きました。

I did not go to school and I went to sensei Yamada. (This day I didn't go to school and I did go to sensei Yamada. I didn't plan to meet Yamada at school, I might have planned to meet him at the park, the konbini e.t.c. I just didn't go to school and I went to see sensei Yamada these two events do not have a logical connection).

私は学校に行かないで山田先生に行きました。

I went to sensei Yamada without going to school.

(This day I didn't go to school and I imply that this might be the proper place to meet him, but I finally managed to see him elsewhere without going to school).