Neither of the definitions you identified is correct for this sentence.
Consider the first, 1a, "to become known." This implies that the two superpowers existed as two superpowers before the war, but only after the war did their existence as superpowers become apparent. I suspect that the actual intention of the sentence is to say that before the war there were more superpowers, or a different set of superpowers, or possibly no superpowers, and after the war there were two superpowers. Therefore, the situation that existed after the war did not "become known"; rather, it came into being.
Consider the second definition you identified, 3, "indicating a state." Look at the examples, which both have the form "emerged X," where X is the "usually good state" being indicated. This formula is, however, absent from the sentence under consideration. If the sentence were "...two superpowers emerged victorious after the Second World War," then definition 3 would apply.
The definition that fits here is 2, "to rise or appear from a hidden or unknown place or condition: to come out into view." I'm not sure why this is listed second, since it is closer to the original meaning of the word, or at least of its Latin source. I would consider the "from a hidden or unknown place or condition" to be superfluous; I would just define "to emerge" as "to come out."
Wiktionary has the fitting definition "to come out of a situation, object or a liquid," with the similar example "The Soviet Union emerged from the ruins of an empire."