I don't want to turn this into a "thing" because Cos, you are actually one of my favorite posters here, no lie. You are thoughtful and reasonable...
But I have to cite the fact that in the Fleischer cartoons, in the George Reeves show, and on and on... They call Supes the Man of Steel. This is his very well known nickname through out pop culture, referenced in many other forms of entertainment, again, FOR DECADES. In songs ranging from pop, rock and hip hop they reference "The Man Of Steel". On tv shows and in film, they've called him the Man Of Steel. That's not my opinion in the least. People in the population at large know that Supes and the Man Of Steel are synonymous with each other, or else they wouldn't be making those references.
As for ole Batsy... I don't think it's a stretch to say that, like it or not, the Adam West show casts a looooonnnggg pop culture shadow. More often than not on that show, in some ways still influential to the public's perception of Batman, they referred to him as the Caped Crusader, even if in comics the preferred moniker was The Dark Knight, especially after Miller's series. The pop culture shift to that moniker has been recent, and mostly due to Nolan's trilogy. I don't think any of that is opinion. I really don't think I am pulling this out my sphincter.