Xperia Z1, Nexus 5 or iPhone 5S? - 1

I'm in the market for a new cell phone in Ontario Canada, replacing my old and tired HTC Amaze 4G. I always believe in getting the best when it is something I hope to use for a long time. The cell technology is confusing and the price range is wide. I don't mind spending a few extra dollars for a better phone. Regarding features, of course, fast and reliable connectivity is important both Wi-Fi and via the cell network. I also travel and exchange the SIM in some countries so it must be unlocked. Being able to use the camera for recording information clearly such as labels etc would be useful. Also, if the pictures were reasonably good, then I would not need my compact camera. I do not want a humongous phone, but I like speed, ease of use and quick access to the features. What do you think? Are you good with the technology and can make a recommendation with the pros and cons. Would greatly appreciate your comments.

Added (1). Thank you Paul for the time you took to explain the benefits of the Nexus 5.It is greatly appreciated.

No question, get the Nexus 5! HERE'S WHY. Among other reasons, if speed is your concern it's the first phone that will feature ultra fast tri-band 4G LTE and greatly improved Wi-Fi speeds. (I've been using it for a few days and it's fast). Moreover, Nexus branded phones are unique because they are the only "Pure Google" devices on the market. It offers the best of both worlds (in my opinion).It isn't cluttered at all, doesn't lag, and is even more fluid to use than the latest iOS platform but also offers all the customization and features Android has to offer. Virtually every other Android smartphone lags (at least some) and seems cluttered/unorganized due to substantial software overlays and bloatware added by manufactures and carriers. This is just a poor attempt to individualize one company's products vs.competitors. For example, the Samsung Galaxy S4 and Note 3 are skinned with "Touch Wiz" and the HTC One uses something called "Sense". Not only do these software skins use up most of your phone's internal memory, they also add so many useless junk that slows the phone down. Apple's OS has always been "stock" and always will be stock - because nobody else manufactures iPhones or designs their software. In other words, it isn't fragmented. A truly stock Android experience was never possible until Google announced the Nexus line of phones and tablets. I won't get into specs and camera, the Nexus 5 is literally right on par with the iPhone 5s in all of that. One thing that sets it apart is the screen quality though. The screen quality on the Nexus 5 is the best I've ever seen on any phone. It makes the Retina display on iOS look like ancient technology. (Not really, but it's definitely a noticeable difference)