MWI is just one of multiple multiverse ideas. Most multiverse ideas (like Tegmark's Levels I, II, and IV) are basically what-if ideas without any direct evidence, but MWI specifically happens to be a more-grounded idea based on trying to make sense of what the (well-tested) Schrodinger equation says about reality.
The first part of your description of MWI ("The many-worlds interpretation of quantum mechanics imagines our universe as one node in an infinitely branching tree of universes where every possible quantum outcome exists in its own universe.") is pretty good, if a slight though common simplification (different branches aren't entirely separate, so envisioning it as a tree is only mostly correct; different branches can sum together or cancel each other out if their configurations are identical).
> trying to make sense of what the (well-tested) Schrodinger equation says about reality.
Nitpick: The Schrödinger equation predicts unitary time evolution (which is another way of saying that physical systems evolve in a deterministic manner). Interpretations of quantum mechanics exist to make sense of the part of quantum mechanics that doesn't follow unitary time evolution, namely the measurement process.
if you read "Our Mathematical Universe: My Quest for the Ultimate Nature of Reality" by Max Tegmark, many worlds falls in the bucket of L3 multiverse. What you're describing is a L4 multiverse (see comments below. seems it's L2, not L4. will leave my mistake in, although i'm sure there is a parallel universe where i did not make this mistake)
Level 2 is universes generally like ours but with different physical constants: https://en.wikipedia.org/wiki/Multiverse#Level_II:_Universes.... Level 4 includes all abstract mathematical structures, like a universe embedded in a Game of Life simulation.
I'm not sure which Sean Carroll talk you saw, but I think he only mentions the differences between worlds in terms of different "choices"-- and these universal constants don't "choose" to be what they are, right?
MWI is just one of multiple multiverse ideas. Most multiverse ideas (like Tegmark's Levels I, II, and IV) are basically what-if ideas without any direct evidence, but MWI specifically happens to be a more-grounded idea based on trying to make sense of what the (well-tested) Schrodinger equation says about reality.
The first part of your description of MWI ("The many-worlds interpretation of quantum mechanics imagines our universe as one node in an infinitely branching tree of universes where every possible quantum outcome exists in its own universe.") is pretty good, if a slight though common simplification (different branches aren't entirely separate, so envisioning it as a tree is only mostly correct; different branches can sum together or cancel each other out if their configurations are identical).