Has Captain Marvel's name changed to "Shazam"?

This was a reply to a comment on Facebook, but since it was so long and thorough for a "Someone is Wrong on the Internet" comment, I figured I would post it here so folks could refer to it with ease. 

Whether or not the name has been "changed" depends on how you look at the character.

In the mid '00's, DC had a series of company-wide epic events, part of which involved Captain Marvel having to take over the job of the old wizard Shazam (who had given Billy Batson the power to speak his name and transform into the hero), and his name changing to "Lord Marvel." Freddy Freeman (Captain Marvel, Jr.) took over the job of Captain Marvel, but after going through the "Trials of Shazam!" 12-issue series (that lasted from 2006-2008,) he became the new hero and was called "Shazam" for a while (Wikipedia says he went back to the name Captain Marvel for a bit).

Then, in 2012, the New 52 came around, and every superhero was "rebooted" or "recreated" and started over. Some of these new versions were drastically different, some were not. The character who was published under the title "Shazam!" in back-up stories n "Justice League," and who later joined the Justice League, was named "Shazam." He had a different costume, different origin, and drastically different powers from the previous characters marketed under that title, all but the Freddy Freeman version having been named "Captain Marvel," and being continuations of, or based on, the original Captain Marvel from Fawcett.

A few years later, DC revealed that the universe in which these "New 52" heroes live is actually part of a "multiverse," and in this multiverse is a world labeled "Earth-5. On this world is a superhero named "Captain Marvel. This world has been said to be "Earth-S," which was the world on which DC placed the original Captain Marvel when they started publishing him in 1972.

To backtrack just a bit...

In 1972, DC began publishing the continuing adventures of the original Captain Marvel, placing him on Earth-S, just as Jay Garrick/Flash was on Earth-2 and Barry Allen/Flash was on Earth1, and while both Earth-1 and Earth-2 had a Clark Kent/Superman, Earth Three had Kal-ll/Ultraman

In 1985-6 DC published the "Crisis on Infinite Earths" saga, which rolled all the alternate Earths into one, rebooting all their characters, re-setting the Big Three of Superman, Batman, and Wonder Woman into the current era, and re-booting Captain Marvel under the title "SHAZAM!" The New Beginning." He had the same powers, a slightly different origin, and the same costume. And while the original Billy Batson and Captain Marvel were separate personalities, now Captain Marvel was simply Billy Batson in the hero's body.

In 1994 DC released Jerry Ordway's re-boot of the character under the title "The Power of SHAZAM!" Again he had a slightly different origin but the same powers, his costume was slightly different, and he maintained the "boy as hero" concept. This was retconned as the part of the new world that emerged from the "Zero Hour" crossover event that happened that same year in DC Comics.

 In 2007, DC released Jeff Smith's "SHAZAM! The Monster Society of Evil" which again re-booted the hero, giving him a slightly different origin and the same costume as the "Power of SHAZAM!" version. In this version the boy and the hero were different personalities, but were starting to merge towards the end

A slightly different version of this character was the hero of the "Billy Batson and the Magic of SHAZAM!" series published for the kid-oriented "Johnny DC" imprint. In this version, the hero had the boy's mind.

Now, if all of these characters, existing on different worlds or after different universe-altering events, count as "different characters," then the New 52 hero, named "Shazam," is not the same character as the original Captain Marvel.

If, however, you want to say that a character with a different origin, different concept, different costume, different powers, a different name, and living on a different world, but marketed under the same trademark as a previous character is the same character as that previous version, then yes, Captain Marvel's name has been changed to Shazam, along with almost everything else about him.