This is one of the most contentious subjects among comic fans, and most particularly, fans of the original Captain Marvel and/or any of his DC variants. There have been Jerry Springer-level fights on the Internet about this. Therefore, in the interest of peace, I offer the flowing breakdown:
First off, the Captain Marvel(s) of whom we are speaking is the one created in 1939 for Fawcett Publications and the various re-boots and re-interpretations committed by DC Comics over the years.
"Shazam" is the magic word that transforms Billy Batson into the hero.
Shazam is also the name of the ancient wizard who gave Billy Batson the power to do this.
In the Fawcett/Earth S/Earth 5 universe (1939-1953, 1972-1986, 2014-2017), his name is Captain Marvel and he has a different personality from Billy Batson. Do you want to know more? And even more?
In the Post-Crisis universe (The New Beginning, 1986-1994), his name is Captain Marvel and he has the mind of Billy Batson. Do you want to know more?
In the Post-Zero Hour universe (Power of SHAZAM! 1994-2012), his name is Captain Marvel and he has the mind of Billy Batson. His look changes when he takes over the Throne of the Wizard Shazam and becomes Lord Marvel. Freddy Freeman then goes through the Trials of Shazam and is called Shazam for a short time, but later is called Captain Marvel. Do you want to know more?
In the Johnny DC universe (Jeff Smith's mini-series The Monster Society of Evil and The Magic of
Shazam series), his name is Captain Marvel. At the beginning of TMSoE he and Billy Batson have separate personalities, but they are combined by the end, and in TMoS he has the mind of Billy Batson.
In Flashpoint (2011) his name is Captain Thunder and he is an amalgam of 6 children, including Billy, but has his own personality.
In the New 52 universe (2012-present), his name is Shazam. He has the hood and crackling logo and a new set of powers. He has the mind of Billy Batson. This is the one in the movie.
When the magic word/name of the wizard is presented like this...
...all caps, with the exclamation point (sometimes depicted as a lightning bolt over a dot), then it is a trademark, and need not necessarily be considered the hero's name.
If Superman of Earth 2, who first appeared in 1938 in Action Comics #1,
Superman of Earth 1, who first appeared in the 1950's,
Ultraman from Earth 3, who first appeared in JLA in 1964,
Post-Crisis Superman, who first appeared in 1986,
Superman from Earth-Prime, who first appeared in the Crisis on Infinite Earths as Superboy,
and the New 52 Superman...
...are all the same character, then yes, the superhero named Shazam is also Captain Marvel, just not in that universe in which he is called Shazam. And he is not called Shazam in any of the other universes.
If you want to consider them different characters because they exist in different universes, then the version of Captain Marvel that exists in the New 52 universe is called Shazam, but he is not the same character.
Done. Stop fighting about it. Please use this for reference.