An odd quirk of many comic book adaptations is that sometimes a character or group will sometimes appear in the adaptation with their name changed for seemingly no reason. Sometimes it's for rights reasons, like when Black Manta was renamed "Devil Ray" in Justice League because Black Manta was set aside for an Aquaman show that never happened. Sometimes it's because of story reasons, such as the latina Gotham detective not being named Montoya in The Dark Knight (because she turned out to be a traitor), and sometimes it's because the adaptation wants to avoid certain words (which is why the Sinister Six was called The Insidious Six in Spider-Man: The Animated Series). It can also be any number of other reasons, but I think it's the last one that's important here today.
Before we get to that, let's talk about the League of Assassins.
The League of Assassins are an organization of killer ninjas that work for Ra's al-Ghul. They first appeared in Detective Comics #405 (September 1970), and have gone on to become a major part of the DC Universe.
But for some reason, the League of Assassins is almost never called that when it shows up in other media. Most famously, in Batman Begins (2005), it's called the League of Shadows. I assume that's because Nolan didn't want Batman to be so blatantly associated with killers. Or maybe "Shadows" just fit the themes of the movie more. Who can say. Regardless, until more recent works like Arrow and Arkham City, if Ra's al-Ghul's gang of ninjas showed up beyond the comics, they were called the League of Shadows, the Society of Shadows, or other names along those lines.
In the comics, the League of Assassins was always the League of Assassins. There was no reason to change it, and they didn't. That is, until Detective Comics #952 (March 2017) (although the storyline, called "League of Shadows", began in the previous issue).
In this story, we learn the League of Shadows not only exists, but it's a sub-group of the League of Assassins created by Ra's al-Ghul that was considered a myth by most until Lady Shiva discovered it. But then Shiva co-opted it (of course) and had it break away from the League of Assassins to accomplish her own goals. The storyline is still ongoing as of this writing, so I don't know what impact or longevity this group will have, but it's interesting to see them introduced!