No Dave, you are completely wrong on this. Here's another source for the actual publication history, note that it's the exact same as I noted above and is found in both the DC and Marvel Captain Marvel Wikipedia pages.
It’s all because of the rights issues, and mainly because how Marvel Comics has the rights to the name Captain Marvel. Fawcett, publishers of the original Captain Marvel/Shazam comics, stopped making them in the ‘50s. In the ‘60s, Marvel Comics trademarked the name Captain Marvel for their own Kree alien superhero, which meant when DC licensed the Fawcett characters in 1972, they had the Fawcett character named Captain Marvel, but couldn’t call the comic Captain Marvel, so they used his transformation cry “Shazam!” for the title.
Again.
1. Fawcett was the publisher of the original Captain Marvel (aka Shazam) from 1939 to 1953.
1.5 A lawesuit file by DC, not by Marvel against Fawcett because Marvel/Shazam was considered to be too similar to Superman. That lawsuit caused the Fawcett's trademark to lapse.
2. Fawcett stopped publication of Captain Marvel in 1953 and there were no Marvel/Shazam comics published between 1953 and 1972 (see 4 below).
3. In 1967 Marvel trademarked the name Captain Marvel and created the Kree military officer Mar Vell. Publication began in 1967.
4. DC licensed Fawcett's characters in 172 and began publishing Marvel/Shazam, but they had to title it "Shazam" because of Marvel's trademark.
I have no idea what you had, but you didn't have any DC published Captain Marvel comics. You might have had some early Shazam comics where they tried to get around the trademark with some clever editing.
WITH ONE MAGIC WORD
SHAZAM!
THE ORIGINAL CAPTAIN MARVEL
They must have heard from Marvel's legal department because within a few issues that bottom line went from "The original Captain Marvel" to "The World's Mightiest Mortal".
Issue #28
>> Because Marvel Comics had by this time established
Captain Marvel as a comic book trademark for
their own character, created and first published in 1967, DC published their book under the name
Shazam![5] Infantino attempted to give the
Shazam! book the subtitle
The Original Captain Marvel, but a
cease and desist letter from Marvel Comics forced them to change the subtitle to
The World's Mightiest Mortal, starting with
Shazam! #15 (December 1974). <<
The guy in green, Mar-Vell the Kree military officer, was introduced in 1967, six years before the first publication of Shazam. And Captain Marvel in the red and blue suit was introduced in '68 or '69. Carol Danvers has been Ms Marvel since 1977 and Captain Marvel since 2012.
Carol Danvers - Wikipedia