Just to be clear, Fuzzibunz is more accurately called a pocket diaper because you have to stuff it with something to become absorbent. When the absorbent material is sewn into the diaper, it's called an All In One (AIO). Occasionally, people call all pocket diapers AIOs or call them pocket AIOs, but in my opinion, it's more clear to keep the names separate.
So, for maximum convenience, you want either pocket diapers or AIO diapers. You have to un-stuff a pocket to wash (which is part of it's beauty: it washes and dries better than true AIOs) and then re-stuff it to use, which some people consider a pain. If you use many different kinds of pocket diapers, you have to keep mental track of which insert goes with which pocket, but that's not particularly hard if you use just 1-3 kinds of pocket, or if you use inserts that fit in any of your pocket diapers.
I love pockets: I think they're the trimmest, driest, and easiest to use after AIOs. Having said that, I don't know much about pockets and newborns. I use only one currently, a Harleyz, but they don't seem to be available anymore. I've heard xs Fuzzibunz are good, but possibly not available anymore? Not sure. So I'm not sure what to recommend for a newborn pocket.
In a size small, the same inserts will probably fit in Fuzzibunz and bumGenius one-size. Happy Heiny (solid colors leak less than print fabrics) is also a really nice fit in size small, but is narrower than a small Fuzzibunz and therefore probably needs it's own insert. Green Acre Designs is not mass produced, but is a great pocket that may be able to use the same insert as a small Happy Heiny. I love Swaddlebees, but they are so narrow that they probably woulnd't work with anyone else's insert, and keeping track of that may be too inconvenient.
Advice and commentary on living overseas
DD1 15, DD2 12, and DS 9