It's not just a relabelling. In collapse theories, the wavefunction stops obeying the Schrodinger equation for a moment and discontinuously jumps to a new state. The times when it performs a discontinuous jump are called "measurements", though this doesn't necessarily mean there's a scientist sitting there with a ruler, it just means that the system has interacted with its environment sufficiently. In the many worlds theory on the other hand, the wavefunction continues to obey the Schrodinger equation for all time, and the natural result of this is that the wavefunction becomes very complicated and entangled, so that the motion of atoms here on Earth is very entangled with photons heading away from us at the speed of light out into deep space, along with pretty much everything else. But there's no mention of "separation" or "worlds" in the basic description of the theory; the one sentence description of many worlds is "the wavefunction obeys the Schrodinger equation all the time with no exceptions".

Where the worlds come in is that it's impossible to do calculations on the wavefunction of the entire universe, so we need to come up with a way of dividing it up into manageable pieces. Not because the theory requires that it be divided into pieces, but because otherwise we couldn't handle the math. The worlds are one of the ways we do that: We break the wavefunction of the universe into approximately perpendicular components that don't interfere with each other very much and don't have too much entanglement making them hard to understand, and we call those worlds. We can further simplify things by just looking at a subsystem of the universe rather than the entire universe, which involves taking a partial trace (this tends to introduce randomness). As time goes on and entropy increases, the entanglement and complexity even within a "world" will continue to increase and at a certain point we may notice, "hey, this component is really complicated now, and it can itself be divided into subcomponents that are approximately orthogonal and don't really interact with each other much, I can simplify my calculations by treating those as separate worlds now". This is what we mean when we say that worlds tend to split apart, but since the worlds are only approximately orthogonal and independent, when you define splitting is really a matter of how much error you're willing to allow in your calculations. (Also, the process of splitting is driven by increasing entropy, so when (if?) the universe reaches a point of total heat death and entropy stops increasing, this will also imply that the worlds have stopped splitting.)

So I'm not sure what you mean by "strong and exceptional". It's just math, and can be compared with experiment just like any other piece of physics. Take the experiments done in the original article. If any kind of collapse had been observed, then that would have straight-up falsified the many worlds theory. Many worlds says that physical systems can become entangled with their environments, but their wavefunctions can never just collapse, and these two cases are distinguishable in a careful experiment. Since collapse wasn't observed when these tests were done, that provides a little bit of evidence in favour of many worlds.

Falsifiability is a little more complicated for collapse theories. People don't agree on the exact definition of a "measurement", and what level of interaction with the environment is required to trigger a collapse, but in order to have a falsifiable theory, it's important that we have a precise, mathematical definition of when a collapse should happen (this definition does not have to be deterministic, it could just give us a probability distribution). So various people have put forward different definitions, and it sounds like these experiments have ruled out a bunch of them, but obviously they haven't ruled out every collapse theory put forwards by every physicist ever. It's a bit like when the LCH failed to find any supersymmetry particles, and some physicists were like, "okay, but in my version of supersymmetry, the particles are heavier than the energies reachable by the LHC so of course we wouldn't expect to have seen them".