Questions about the original iOS as well as all iOS in general can be tagged as iOS.
The underlying structure of Mac OS X was adopted by iOS, however the implementation provides a fairly radically different user experience than OS X. Over time, features and design idioms are cross pollinating with many features from iOS finding their way to OSX in versions Lion and the upcoming Mountain Lion release.
The term iOS came about after the iPhone was originally released once it became clear that iPhone OS and iPod Touch OS were too unwieldy. The iPad and second version of Apple TV (without an internal hard drive) all run on the same core OS.
Each device gets a customized build even when the major build numbers (e.g. iOS 5.0.1) are the same and the software is released and documented as one build. The current as of 3/7/2012 version is 5.1.