There are at least ten different conceptions of how the World can be made of many worlds.

But are those just definitional disputes? Or are they separate claims that can be evaluated. If they are distinct, in virtue of what are they distinct. Finally, do we have good grounds to care (morally) about those fine distinctions?

Max Tegmark's taxonomy is well known here. 

Brian Greene's is less, and has 9, instead of four, kinds of multiverse, I'll risk conflating the Tegmark ones that are superclasses of these, feel free to correct me:

In his book, Greene discussed nine types of parallel universes:

• (Tegmark 1) The quilted multiverse only works in an infinite universe. With an infinite amount of space, every possible event will occur an infinite amount of times. However, the speed of light prevents us from being aware of these other identical areas.
• (Tegmarks 1 and 2) The inflationary multiverse is composed of various pockets where inflaton fields collapse and form new universes.
• The brane multiverse follows from M-theory and states that each universe is a 3-dimensional brane that exists with many others. Particles are bound to their respective branes except for gravity.
• The cyclic multiverse has multiple branes (each a universe) that collided, causing Big Bangs. The universes bounce back and pass through time, until they are pulled back together and collided again, destroying the old contents and creating them anew.
• (Tegmarks 2) The landscape multiverse relies on string theory's Calabi-Yau shapes. Quantum fluctuations drop the shapes to a lower energy level, creating a pocket with a different set of laws from the surrounding space.
• (Tegmarks 3) The quantum multiverse creates a new universe when a diversion in events occurs, as in the many-worlds interpretation of quantum mechanics.
• The holographic multiverse is derived from the theory that the surface area of a space can simulate the volume of the region.
• (Related to Bostrom's Simulation Hypothesis) The simulated multiverse exists on complex computer systems that simulate entire universes. (for the sake of brevity I'll consider dust theory to be a subset of this)
• (Tegmark's 4) The ultimate multiverse contains every mathematically possible universe under different laws of physics.

I don't understand branes well enough (or at all) to classify the others. The holographic one seems compatible with a multitude, if not all, previous ones. 

Besides all those there is David Lewis's Possible Worlds in which all possible worlds exist (in whichever sense the word exist can be significantly applied, if any). For Lewis, when we call our World the Actual World, we think we mean the only one that is there, but what we mean is "the one to which we happen to belong".  Notice it is distinct from the Mathematical/Ultimate in that there may be properties of non-mathematical kind. 

So Actual_lewis= Our world  and Actual_{most everyone else}=Those that obtain, exist, or are real. 

The trouble with existence, or reality, is that it is hard to pin down what it is pointing at. Eliezer writes:

The collection of hypothetical mathematical thingies that can be described logically (in terms of relational rules with consistent solutions) looks vastly larger than the collection of causal universes with locally determined, acyclically ordered events. Most mathematical objects aren't like that. When you say, "We live in a causal universe", a universe that can be computed in-order using local and directional rules of determination, you're vastly narrowing down the possibilities relative to all of Math-space.

So it's rather suggestive that we find ourselves in a causal universe rather than a logical universe - it suggests that not all mathematical objects can be real, and the sort of thingies that can be real and have people in them are constrained to somewhere in the vicinity of 'causal universes'. That you can't have consciousness without computing an agent made of causes and effects, or maybe something can't be real at all unless it's a fabric of cause and effect. It suggests that if there is a Tegmark Level IV multiverse, it isn't "all logical universes" but "all causal universes".

and elsewhere

More generally, for me to expect your beliefs to correlate with reality, I have to either think that reality is the cause of your beliefs, expect your beliefs to alter reality, or believe that some third factor is influencing both of them.

Now another interesting way of looking at existence or reality is  

Reality=I should care about what takes place there

It is interesting because it is what is residually left after you abandon the all too stringent standard of "causally connected to me", which would leave few or none of the above, and cut the party short.  

So Existence_yud  and Existence_{moral-concern} are very different. Reality-fluid, or Measure, in quantum universes is also different, and sometimes described by some as the quantity of existence. Notice though that the Measure is always a ratio - say these universes here are 30% of the successors of that universe, the other 70% are those other ones - not an absolute quantity.

Which of the 10 kinds of multiverses, besides our own, have Existence_yud  Existence_{moral-concern} and which can be split up in reality-fluid ratios?

That is left as an exercise, since I am very confused by the whole thing...