Rights/trademarks is immaterial given that DC (and by extension New Line, who are using a DC set of rights) have a 1973 settlement with Marvel Comics agreeing never to market the character that way. Which is why for example all the show they ever made were marketed as "Shazam", not "Captain Marvel".
It's true they used the Captain Marvel name within some of those shows (but interestingly WB's most recent cartoon, from 2013, used the Shazam name throughout, and this is probably the best indicator of WB (and New Line, which is really just a banner there now) being serious about the name change.
More to the point the majority of those shows - with the exception of 2009's Superman/Shazam animated movie - were made before Marvel Studios was a success (prior to 2008, when there had been Marvel Studios, but these were different legal entities); everything since has used the Shazam name which has been current in the comics since New52, seven years ago.
Since it's questionable whether most casual moviegoers would even care about comic book origins of characters but would certainly know that Marvel is a studio and Captain Marvel (who, naturally, belongs to Marvel Studios) has a movie coming out, the idea that a major studio group like WB would permit a hundred million dollar investment to be risked over a stupid name argument is laughable.
Even a joke about it making it into the trailer (and remember, trailers are produced out of house by specialists, and often released before anybody on the main production crew has seen them) would kill the Shazam movie dead, because of that very clear 1973 settlement which DC and WB have up until now abided by. Changing the guy's name in New52 wasn't an accident; it was corporate-mandated synergy.
Consider yourself lucky they didn't change it to Grindr.>>99354864
The Wizard is only listed as "The Wizard" so far. Shazam is the title character's name.