Dbat: reading on an electronic device shouldn't count as "screen time." And that doesn't just mean e-readers, it means reading a newspaper on the web, or browsing Wikipedia, or anything else where you're basically just reading. Your brain is processing written text, which is very different from interacting with the three-dimensional world, but which our culture has agreed is extremely important and worth the time investment.
A caveat to this, which I don't believe is applicable to e-readers: there is convincing research showing that the strong blue-spectrum light emitted by screens disrupts sleep cycles. Even if they're reading text, I would keep kids (and I try to keep myself) away from computer screens in the hour or two before bed. Eye strain can also be an issue.
Speaking personally, I don't find that reading online has the same immersive and relaxing effect as reading a book. It's too jumpy and manic, with hyperlinks and so on. Ereaders count as books the way I use them, though (which is...to read books only).
