I came across a post about this and thought I'd make a similar thread. Really for me, it all comes down to which is really the most "essential" part of the title.
Well, in the first issue of TMNT, Splinter recounts the origin of his master Hamato Yoshi, rival Oroku Nagi, and love interest Tang Shen. Two of these men were apart of the Foot Clan, the most powerful Ninja clan in Japan. But their bitterness over winning a woman's heart led to Nagi beating the shit out of Shen; only to be caught at the last second by Yoshi as he furiously pummels Nagi off-panel. Only blood remains on his fists. By dishonoring his clan, his two choices were to flee to another country and start a new life or commit sepekku. Choosing the former, they both settle down for a while only to be found and killed by Oroku Saki years later.
Splinter was left alone scavenging the trash bags when he saw a blind man walking down the street; not noticing the speeding truck coming, a kid knocks the old man to the other side as a canister falls out. Bouncing until it hits a kid's terrarium, dropping down into the sewer drain below. You know what's after.
So, from all that, I think we can infer that the "Ninja" is principal to their characteristics, the "Turtle" describes the characters, and the "Mutant" explains their appearance. That just then leaves the "Teenage" part of the equation which is only a temporary part of the stories until at least Volume 3. Except, it seems like the Turtles are being pigeonholed on exactly the "teenage" font since the buy out. Outside Michaelangelo from the Last Ronin, Paramount and I fear even IDW have made it to where the Turtles can never officially aged. Kinda like what's happened to Spider-Man as well but I'm getting ahead of myself.
Tldr; which is essential or appealing from the title itself?