Maybe we can find some common ground to start with.
The central idea of the MW and similar interpretations is that wave
function collapse does not occur. Instead, the wave function is universal
and evolves in time strictly by means of the Schrodinger equation.
This is the mathematical theory which supposedly describes our universe.
Now we have to try to understand how or whether this theory actually
could apply to our universe. This gets into the issue of mapping, which
I talked about yesterday. Maybe in theory the mathematical formula is
enough and whatever evolves from it gets to experience it from the inside.
But in practice, to try to judge whether a theory matches our universe,
we need to have a way to map between its predictions and what we see.
With MW it is especially difficult since we observe what seems to be wave
function collapse during measurement interactions. Therefore the thrust
of the discussion is how to reconcile the appearance of wave function
collapse with the (theoretical) reality that it does not collapse. The
focus is on measurement-like interactions and how it could be that an
observer would seem to see collapse.
The actual definition of what constitutes a measurement is not looked
at too closely in this analysis. Instead, as with the conventional
interpretation, we assume that we have classical-like measurements and
explore how to relate those to the reality of what is going on with the
wave function. That leads naturally to the notion of decoherent branches
of the wave function, and hence the appearance of "many worlds".
It sounds like what you want to do is to explore the issues of what
constitutes a measurement, and when decoherence occurs. This would then
lead to a more detailed understanding of when worlds can be said to split.
This topic is discussed in conventional measurement theory, from a
different point of view. There it is shown that as a quantum phenomenon
undergoes amplification to large numbers of particles, the nature of
the wave function changes. It changes from a superposition of states
to a mixture of decoherent states. This can be derived purely with
the continuous evolution of the wave function, without introducing any
notions of wave function collapse.
In the context of the MWI, once you have a mixture of decoherent states,
we can interpret this as being different branches of the wave function
which no longer will interfere with each other, thus giving the effect
of "many worlds".