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I always thought a transaction can be accepted when all its inputs are correctly provided with their unlock script (which usually contains a signature) ——yes transactions may also be rejected with output > input etc but that’s not the topic

Does the transaction itself also need to be signed? If yes, when I am having inputs with different unlock scripts, which one should I use to sign the transaction?

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Transaction do not need to be signed. They are if the locking script requires so, which is the most usual thing, but scripts redeemable without a signature can also be valid.

That being said, a transaction redeeming from a script that does not requires a signature could be highly insecure, since a peer (or a miner) that receives so can easily change the output.

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