It's that part of the wavefunction.
That's no good if I'm trying to evaluate MWI as a physical theory. Those are just words. You can try to extract a "branch" from a wavefunction or a state vector in a variety of ways. You might focus on a specific classical configuration (then we might have to quibble about whether you mean the point in configuration space that is labeled by that configuration, or whether you mean the delta function peaked at that point in configuration space). Or we might be talking about some other sort of basis function in terms of which a wavefunction might be decomposed (momentum eigenfunctions, wavelets, eigenfunctions of other observables). We might be talking about reduced density matrices rather than pure states...
If someone wants to insist that, in their version of MWI, there is a locally specific reality only for the region of the observer's brain corresponding to conscious observation, they can dodge some of the details by saying we don't know enough neuroscience yet. But they should be able to make precise statements about what sort of mathematical vivisection is going to be performed on the local wavefunction in order to get the "part of the wavefunction... observing stuff". Otherwise, they don't yet have a physical theory, just a vague hope that some version of Many Minds will work out.
And by the way, if we are not saying that branches correspond to universe-wide configurations, but rather just to local configurations, what's responsible for tying together a million different information-processing events into a single mind-state? Every neighboring molecule in the brain ought to be locally branching, constantly, and independently of branchings on the other side of the skull. How could you ever justify speaking about "the" state of someone's brain, even just as one branch of an Everett multiverse? The cognitive state will have to "supervene" on something which, from an atomic and subatomic level, is already a very intricate superposition.
That's no good if I'm trying to evaluate MWI as a physical theory. Those are just words.
No, you have this backwards. You're looking for some words. I'm telling you there are just numbers. The numbers are all we need to be evaluating here.
The subject has already been raised in this thread, but in a clumsy fashion. So here is a fresh new thread, where we can discuss, calmly and objectively, the pros and cons of the "Oxford" version of the Many Worlds interpretation of quantum mechanics.
This version of MWI is distinguished by two propositions. First, there is no definite number of "worlds" or "branches". They have a fuzzy, vague, approximate, definition-dependent existence. Second, the probability law of quantum mechanics (the Born rule) is to be obtained, not by counting the frequencies of events in the multiverse, but by an analysis of rational behavior in the multiverse. Normally, a prescription for rational behavior is obtained by maximizing expected utility, a quantity which is calculated by averaging "probability x utility" for each possible outcome of an action. In the Oxford school's "decision-theoretic" derivation of the Born rule, we somehow start with a ranking of actions that is deemed rational, then we "divide out" by the utilities, and obtain probabilities that were implicit in the original ranking.
I reject the two propositions. "Worlds" or "branches" can't be vague if they are to correspond to observed reality, because vagueness results from an object being dependent on observer definition, and the local portion of reality does not owe its existence to how we define anything; and the upside-down decision-theoretic derivation, if it ever works, must implicitly smuggle in the premises of probability theory in order to obtain its original rationality ranking.
Some references:
"Decoherence and Ontology: or, How I Learned to Stop Worrying and Love FAPP" by David Wallace. In this paper, Wallace says, for example, that the question "how many branches are there?" "does not... make sense", that the question "how many branches are there in which it is sunny?" is "a question which has no answer", "it is a non-question to ask how many [worlds]", etc.
"Quantum Probability from Decision Theory?" by Barnum et al. This is a rebuttal of the original argument (due to David Deutsch) that the Born rule can be justified by an analysis of multiverse rationality.