「ないで」と「なくて」の違い

I was learning てほしい grammar from genki. I realized I have seen two te-forms for verb short negative forms.
The example provided was

私は友達にタバコをすわないでほしいです

Is it okay if I say

私は友達にタバコをすわなくてほしいです

How do I know which form to use? Are these okay?

昨日勉強しなくて今日の試験に失敗しました。(Because I did not study yesterday, I failed todays exam)
今日雨が降らなくても出かけません。(I wont go out, even if it doesn’t rain)

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Auxiliaries like ほしい and ください attach to ないで and not なくて

Shedding some light on this, ないで’s primary use is to give negative verbs an adverbial use.

走らないで部屋に入る enter the room without running
見ないで予想してみる guess without looking

Using なくて is more like a typical て sentence, which can have various uses.

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That’s a new way to look at it for me, and makes a lot of sense! Thanks!

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Wow. Nice explanation.
教えてくれてありがとう