If you have dynamo-powered bicycle lights, you already own an electric-powered bicycle! Consider: as you pump your legs up and down on the pedals, you make the wheels rotate. A small dynamo (generator) mounted on the rear wheel produces a tiny current of electricity that keeps your back safety lamp lit in the dark. Now suppose you could run this process backward. What if you removed the lamp and replaced it with a large battery. The battery would kick out a steady electric current, driving the dynamo in reverse so that it spun around like an electric motor. As the dynamo/motor turned, it would rotate the tire and make the bike go along without any help from your pedaling. Hey presto: an electric bike! It may sound a bit far-fetched, but this is more or less exactly how electric bikes work.
Both the Stromer and the Stoeckli are very nice looking designs and easy to handle. Reliability for both does not seem to be up to Swiss standards, e.g. some of the 2012 Stöcklis seem to have bad contacts. However, as of 2013, most of these problems should be fixed. The Cube got criticized for its battery/saddle system, but this has probably been fixed in more recent edition and it has less "punch" then the other's since the motor is smaller (324W?). I don't know about the BH Neo Nitro. Given the relatively low price of the BH Nitro, it may be the best buy in this category if you plan to cover smaller distances (the battery is limited to 9Ah, and Spain's industry does need some help ;) Anyhow, all of these models come with a variety of motors and country-specific modifications. E.g. in France, the BH Nitro comes with a 350W motor and is electronically limited to 25km/h, whereas in Switzerland you either can get a 500W - 45 km/h version or a 250W? - 25km/h version.
The oldest patent for an electric bike I've been able to find at the US Patent and Trademark Office is this one, by Ogden Bolton, Jr. of Canton Ohio, which was filed in September 1895 and granted three months later. You can see from these original diagrams that it bears an amazingly close resemblance to modern electric bikes. In the general picture on the left, you can see there's a hub motor on the rear wheel (blue), a battery suspended from the frame (red), and a simple handlebar control to make the thing stop and go. In the more detailed cutaway of the hub motor on the right, you can see there's a six-pole magnet in the center (orange) bolted to the frame and an armature (made from coiled wire, yellow) that rotates around it when the current is switched on. It's quite a hefty motor even by modern standards; Ogdon mentions "a heavy current at low voltage—for instance, to carry one hundred amperes at ten volts." So that's 1000 watts, which is about twice the power of a typical modern bike hub motor.
To qualify as an electric-assisted bicycle under state law they need to have a seat and fully operable pedals for human propulsion, meet federal motor vehicle safety standards, an electric motor that has a power output of not more than 1,000 watts, maximum speed of not more than 20 mph (electric motor and human power combined), disengages or ceases to function when the vehicle’s brakes are applied, two or three wheels