What about the sentence before the bolded one? It makes no distinction between positive and negative claims regarding the burden assumed by the one making the claim.
Actually, it does quite clearly say that the burden of proof is on the person asserting the claim.
And this shows a distinction between a positive and a negative claim how? I can just as easily assert a negative claim as I can assert a positive claim.
Specifically definition 1
To declare with assurance or plainly and strongly; to state positively.
he would often assert his beliefs to us
I can know for a fact that there is no elephant in my backyard.
That is me making a negative claim, declaring positively, that there is no elephant in m backyard.
We also have this a few lines down from the link that Tom posted
Proving a negativeWhen the assertion to prove is a negative claim, the burden takes the form of a negative proof, proof of impossibility, or mere evidence of absence.
If this negative assertion is in response to a claim made by another party in a debate, asserting the falsehood of the positive claim shifts the burden of proof from the party making the first claim to the one asserting its falsehood, as the agnostic position that "I don't believe that X is true" is different to the explicit denial "I believe that X is false".[8]
So please take a look at the bolded section.
This is exactly what has happened in the other thread.