~てほしくない, ~なくてほしい. Can you use them interchangeably?

I was doing one of the sentences in the N4 section to prepare for the test this summer, and I got it wrong due to an incorrect conjugation. It had me thinking, however, is there a difference between putting ほしい into the negative case and putting the verb into the negative て form and leavning ほしい as it is? I will provide the sentence below:

ブラジル____ (負ける)(don’t want (you) to lose like Brazil)

My answer: ブラジルのように負けなくてほしい。X (marked as incorrect)

Correct answer: ブラジルのように負けてほしくない。

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Isn’t the difference the same in English?

I don’t want you to lose like Brasil.
( = ほしくない )

Vs

I want you not to lose like Brasil.
( = ないでほしい )

--- edit ---

as @nekoyama pointed out below, it should be ないで for the second example.
Not なくて as I mistakenly put.

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@rikvg
That’s what I was thinking, but I wasn’t sure if one sounded more natural/less natural in Japanese. Maybe that’s why the system marked it wrong?

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You can make the verb negative, but you should use ないで instead of なくて (this applies to other ways to make requests too, not just ほしい).

And yes the difference is similar to English. If ほしい is negated, it’s a much softer statement; you don’t particularly want someone to do something. If you negate the verb, you actively want someone to not do something, for example 誰にも言わないでほしい.

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