tl;dr: It was actually a combination of both. Before Crisis on Infinite Earths there were two Superman characters. Afterward, one has died and the history of the other had been rewritten. Man of Steel tells the new story as it was post-Crisis.
The Crisis on Infinite Earths storyline was an incredibly complex mess, meant to clean up the even bigger mess that had been made of DC's continuity at that point. Over the decades, Superman's backstory and powers had been constantly changed and evolved (he originally couldn't fly; he was originally the only Kryptonian but later found Supergirl, etc.)
By the time of the crisis event, there were already two Superman characters in DC history. The first was "Kal-L" from the Golden Age, the original Superman from 1938. The other was "Kal-El" from the Silver Age, who was introduced as Superboy in 1945. When Superboy's backstory conflicted with Kal-L's backstory, DC introduced the concept of the "multiverse" to explain these different versions of the characters -- young Kal-El (from "Earth-One") even meets young Kal-L (from "Earth-Two") at one point and trains him.
On top of that, most of the other DC characters had run into similar problems, with two (or more) versions of the same character appearing in different versions of Earth, sometimes crossing over to interact with each other. This got too confusing for anyone to follow, and if you were trying to read multiple series with the same characters, it was a big headache. So, DC came up with the Crisis.
During the Crisis, a super-powerful multi-dimensional being started messing around with the multi-verse. All the heroes from Earth-One and Earth-Two came together to fight him. In the end, the older Earth-Two Superman sacrificed his life, and Kal-El from Earth-One was left alive. All the different Earths were merged together into a single universe, and everyone's backstories have been rewritten into new versions of the character.
Man of Steel lays out the new, revised, post-Crisis version of Superman's origin.