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1. We want John to win.

2. We want for John to win.

3. We prefer for John to win.

4. We prefer that John will win.

5. We believe there to be a cellist in the band.

6. We believe that there is a cellist in the band.

There are some verbs that have many different forms but have the same meaning.

1, 2 and 3, 4 are grammatical according to my book.

Q) 1, 2 and 3, 4 and 5, 6 are used without change of the meaning?

and what are more likely to use in conversation?
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These sound unremarkably acceptable to me:

1. We want John to win.

4. We prefer that John win.

5. We believe there to be a cellist in the band.

6. We believe that there is a cellist in the band.

I think that your book is not a reputable reference.
Comments  
Teachers: We supply a list of EFL job vacancies
1. We believe John to be interested in getting the job.

2. We believe for John to be interested in getting the job.

Does #2 sound OK too?
No, #2 does not work at all. 'Believe for' does not collocate.