There is no arbitrary burden of proof on anyone. It all comes down to what actions are going to result from the claim, and what do the actors who will enact those actions expect before they determine what act, if any, should be taken.

It plays a role in the court of law because the decision can have a huge impact on people's lives.

If I was just making a claim that Andromeda has a serious unicorn problem and that they need our thoughts and prayers before they are overrun entirely by marauding horny horses, it doesn't come with any obligation to convince anyone, even though it is entirely true and could be easily proven if I had more space in the margins.

For defamation, it gets more complicated. Some jurisdictions require the defendant prove what they said, others require the plaintiff to prove it false. But the main thing with both is that there is a burden of proof because there's consequences directly tied to whether the claim is true.