Even though Ezra Miller's Barry Allen already saved the world in Zack Snyder's Justice League, he isn't officially called "The Flash" yet. After his small appearances in Batman v Superman: Dawn of Justice and 2016's Suicide Squad, Barry Allen finally got his chance to shine alongside his fellow Justice League founding members in the Snyder Cut. When fighting Steppenwolf, Barry was able to tap into the Speed Force to rewind time a few moments and save the Justice League. But despite his impressive efforts, he hasn't yet earned his official superhero moniker.
Superman has been so popular (and controversial) in the DCEU that the whole world knows his iconic name. This is also the case with the infamous Gotham vigilante known as Batman, who remained an urban myth for decades, and with Arthur Curry Jr. aka Aquaman, who also became a living legend before the Justice League assembled for the first time. The Justice League Snyder Cut marks the first mention of the name "Cyborg" when Diana Prince describes Victor Stone, as well as the first mention of the name "Wonder Woman" - curiously, from an awestruck Barry Allen.
Barry Allen only gets his "Flash" title during the Arrowverse's Crisis on Infinite Earths crossover, which reveals that the DCEU's Barry Allen gets his first experience with different realities when he first meets a multiverse version of himself - Grant Gustin's Flash. During their brief encounter, this seasoned Barry Allen says, "I'm also The Flash", causing the DCEU's Barry Allen to have an epiphany about the name. Even though he already fought alongside the Justice League and discussed the idea of the multiverse with Cyborg, he's still a novice in superhero matters at this point.
The Snyder Cut confirms that Barry Allen has done much more than just pushing people and running away. He's a hero who's skilled enough to save Iris West from a car crash and rewind time to save his teammates without taking even credit for it. However, it's also his speed that keeps him from becoming famous. Nobody but Iris knew that he saved her, security cameras don't capture his heroic actions, and not even the Justice League ever realized that he saved them from certain doom.
Barry's Speed Force lighting is the only sign that a certain entity was present for an instant, but witnesses have only seen a flash of lightning with nobody attached to it. This would be a good reason for the general public to start calling him "The Flash" or for Lex Luthor or Amanda Waller to come up with the same codename, but that apparently isn't the case. While Barry will probably get an official superhero name once he faces a big threat by himself, the fact that he hears "The Flash" for the first time through DC's multiverse might be an important plot point in Andy Muschietti's The Flash. Zack Snyder's Justice League serves as Barry Allen's first major adventure as a hero, but his official name will likely come in his first solo story.
