My understanding is that, whatever version of Copenhagen you adhere to, you no longer believe that there is another version of the scientist somewhere who has measured 1.

Depends. From the outside observer's point of view, he can be in a superposition of [has measured 0] and [has measured 1]. From the scientist's point of view, the collapse has happened.

...but then I have absolutely no idea what distinguishes Copenhagen from many worlds.

That's why it's called "interpretation". It's the way how we speak about it, and some untestable statements about consciousness perhaps with some philosophical implications, which make the whole difference. Of course, an objective collapse is a different thing, but I don't believe much Copenhagenists today believe in an objective collapse.

According to many worlds, there are multiple versions of earth---one in which the scientist observed 0, and one in which the scientist observed 1.

Such statement can be misleading. There is one version of Earth, but the individual observers see only certain projections. The difference between MWI and single-world interpretations is that MWI says that all projections are experienced.