DagoodS's comment in Atheism and the burden of proof illustrates an important distinction between the related but different concepts that I'll label as the more-or-less static "global" burden of proof and the ever-shifting "local" burden of proof.
In a legal case, the prosecution and the defense have substantively different tasks: the prosecution (or plaintiff in a civil case) has to prove the specific charge. The defense, however, never has to prove the opposite of the specific charge; the defense (absent an affirmative defense) must merely undermine the prosecution's case. This is, as best I understand it, the point DagoodS is making about the technical legal definition of the "burden of proof"; we can see this as the global burden of proof.
There is also, even in a legal case with a static global burden of proof, a de facto shifting local burden of proof. Once the prosecution introduces some evidence supporting the specific charge, if the defense does nothing, the jury might conclude that because the defense has not raised any doubt at trial no doubt actually exists. The defense — if they wish to prevail — now has the burden of raising doubt. This burden is for all practical purposes, a burden of proof, since the defense must give a reason to raise doubt. It's in some sense a "lesser" burden, since the defense does not need to prove the opposite of the overall charge; they need only prove the lesser point that the prosecution's evidence is not as probative as the prosecution would have the jury believe.
The prosecution might have to prove, for example, that the defendant was near the scene of the crime. Because they have a global burden of proof, they cannot merely allege that the defendant might have been near the scene of the crime, they have to introduce actual evidence proving that the defendant actually was at the scene. The defense never has to prove that the defendant was not at the scene of the crime, they merely have to cast doubt on the prosecution's case: they can in effect prove only that the defendant might have been elsewhere. The prosecution has the global burden of proof.
But the prosecution might bring in an eyewitness who will testify that he saw the defendant near the scene of the crime. At this point, the defense has a local burden of proof. They cannot merely allege that the witness might have been mistaken; they must introduce actual evidence to prove that the witness was mistaken (or at least unreliable enough to raise a reasonable doubt). The defense has a local burden of proof.
The point of global and local burdens of proof is that the burden is on one party to perform a specific kind of task, not on each party to perform the same kind of task more effectively than the other.