Undefined behaviors allow compilers to generate faster code in some cases. Consider two different processor architectures that ADD differently:
Processor A inherently discards the carry bit upon overflow, while processor B generates an error. (Of course, Processor C inherently generates Nasal Demons - its just the easiest way to discharge that extra bit of energy in a snot-powered nanobot...)
If the standard required that an error be generated, then all code compiled for processor A would basically be forced to include additional instructions, to perform some sort of check for overflow, and if so, generate an error. This would result in slower code, even if the developer know that they were only going to end up adding small numbers.
Undefined behavior sacrifices portability for speed. By allowing 'anything' to happen, the compiler can avoid writing safety-checks for situations that will never occur. (Or, you know... they might.)
Additionally, when a programmer knows exactly what an undefined behavior will actually cause in their given environment, they are free to exploit that knowledge to gain additional performance.
If you want to ensure that your code behaves exactly the same on all platforms, you need to ensure that no 'undefined behavior' ever occurs - however, this may not be your goal.
Edit: (In respons to OPs edit)
Implementation Defined behavior would require the consistent generation of nasal demons. Undefined behavior allows the sporadic generation of nasal demons.
That's where the advantage that undefined behavior has over implementation specific behavior appears. Consider that extra code may be needed to avoid inconsistent behavior on a particular system. In these cases, undefined behavior allows greater speed.