New Zealand Law
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That is an unusual interpretation of employment law by your manager. You are entitled to give notice and be paid out, and to say you have to pay them is nonsense. Often in this situation you do have to leave right away, and take what is called gardening leave. But you are entitled to the pay, although if you are starting right away and the previous employer gets difficult you will at least not lose income. They must also pay holiday pay
If they insist on you working out your leave, that would be unusual, and unlikely. If that was the case however you would not be asked to take gardening leave. If you get a start date from the new employer then leave and join them. If your former employer gets silly about payment you can ask a Labour Inspector to help get your wages which is a free service, and you could also claim a personal grievance for the shabby treatment. People change jobs all the time and they just need to get used to the idea, as they do not own you.
They cannot deduct the payment from your holiday pay, and if they try, this would also be a matter for the Labour Inspector and a personal grievance claim
That may be contrary to the Holidays Act. But if you give notice, and they ask you to leave without working out the notice, then that is not what the clause contemplated, and would then count as a dismissal and entitle you to claim damages, if they then tried to deduct your holiday pay.