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Manfred comments on A note on the description complexity of physical theories - Less Wrong
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Yes, it does. The Copenhagen interpretation says that when you observe the universe, your observation becomes right, and your model of the world should make a 100% certain retrodiction about what just happened. This is mathematically equivalent to letting the a MWI modeler know which world (or set of worlds with the same eigenstate of the observable) they're in at some time.
However, in Copenhagen, the universe you observe is all there "is." If I observe the electron with spin up, there is no other me that observes it with spin down. The probabilities in Copenhagen are more Bayesian than frequentist. Meanwhile in MWI the probabilities are frequencies of "actual" people measuring the electron. But since there is no such thing as an outside observer of the universe (that's the point), the difference here doesn't necessarily mean this isn't an argument about definitions. :P
Your Copenhagen Interpretation looks like starting with Many Worlds, and then rejecting the implied invisible worlds as an additional assumption about reality.
My Copenhagen interpretation (the one I use to demonstrate ideas about the Copenhagen interpretation, not necessarily the interpretation I use when thinking about problems) looks like the Copenhagen Interpretation. And yes, it is close to what you said. But it's not quite that simple, since all the math is preserved because of stuff like entanglement.