Everything in one book I'd say. For me, that means both powered and unpowered flying.
As others have said, there's a requirement to log the hours in some detail, but there's no hard and strict rule about where and how to log it. As long as you are able to reproduce your flying history in sufficient detail, anything goes.
And let's be honest. Some of the experience you gain while flying on your FAA license will count towards currency on your JAA license, or the other way around. By having the record of that experience spread out over two books, with one of those sitting at home on a different continent, you might have a hard time proving that experience.
For instance your "three landings in the last 90 days" is not specifically restricted to a license/country/aircraft registration. (Although it might be restricted to an airplane class/type - I'm not quite sure about that one.)
Saves having scribbles in your pro log book for interviews.
I wonder if that is a bad thing. I'm a neat guy and even make sure all my logbook entries are in black pen, not in another color. But I have also heard of people who really "personalize" their logbook, using up multiple lines to record noteworthy flights, adding photos and other memorabilia. As long as it doesn't become too 13-year-old-girlish, at least it gives you something more to talk about during the airline interview. And it shows that you've actually enjoyed flying different types in different circumstances, rather than just having done hour-building for the sake of getting the hours in the logbook.