The four main animated series of TMNT have had at least one theme song that gives the turtles one sentence descriptions. So I'm going to compare them (in order of age each time, 87, 03, 12, Rise).
Leonardo leads - This one's to the point. He does lead. Doesn't really say anything else about him, but it tells us his role in the team.
Leonardo's always in control - This still tells us that Leonardo's in charge but also implies self-control and a personality that wants to keep things under control, rather than just telling us his role on the team.
Leonardo's the leader in blue, does anything it takes to get his ninjas through - 12 gives us team role again, but the second bit does give us more of his personality. It implies a certain level of ruthlessness and I do find 12 Leo to be the most ruthless Leo.
Leo's making a scene - Sorry, buddy, you're not the leader and your personality is just loud. This fits for a Leo whose storyline is a great deal about finding his role and what he's good at.
Donatello does machines - As a friend on discord pointed out this doesn't (especially combined with the visual) tell us he does machines well. Donatello's machines are sometimes more trouble than they're worth, but with the amount of mad scientists these turtles go up against they very much need someone doing machines on their team, and he does brilliantly even more often than he caused the problem in the first place. Like Leonardo this mostly tells us his team role.
Donatello's the brains of the bunch - 03 is the only one that doesn't mention machines, the 03 theme song moves towards personality for all of them rather than team role for some and personality for others. So Donatello gets mentioned as the smart guy rather than the one who makes machines. It's also true of him in more ways than of some other Donatellos - this one's smart about things other than science, too.
Donatello's the fellow has a way with machines - He's the machine guy again, but this one's more complimentary about it.
Donnie rocks machines - I love how these get steadily more complimentary about how good Donatello is with machines even while mostly sticking to machines as his thing.
Raphael is cool but rude - 87 gives two their team roles and two their personalities in the theme song, but then Raphael and Michelangelo have much less fixed team roles. I feel like just announcing a character is cool up front wouldn't be done in modern cartoons, too. Is he cool? Isn't that up to the viewers? But it definitely tells us how Raphael sees himself.
Count on Raphael to throw the first punch - 03 once again neatly combining team role (fighter) with personality (a bit aggressive and impulsive).
Raphael's got the most attitude on the team - We're back to what 87 says in new phrasing, with less just announcing he's cool. 12 Raphael is also the one who throws the first punch, mind you, but that's not what his theme song highlights.
Raph's ready for a fight - Rise is following 03's lead on this one. Rise probably has the least attitude of any Raphael, not least because he is the authority to his brothers so who would he be rebelling against?
Michelangelo is a party dude - He is! He's a lot of other things too, and I think this one is actually a little sad when you think about how many parties a mutant turtle is likely to be invited to, but it conveys his carefree attitude and desire to have fun in the moment. And he also does love parties and throws birthday parties for everyone including Irma.
The wise guy is Michelangelo - I think this Michelangelo is the only wise guy Mikey. The role goes to Raphael in 87, Leo in Rise, and no one really in 12. It also singles him out a bit, the others in 03 have all been given a sentence that's mostly about their personality but also alludes to their role on the team, he's the only one who has no definite role to allude to.
Michelangelo, he's one of a kind, and you know just where to find him when it's party time - Like 87 he likes parties and we're also highlighting that he's weird, I guess.
Mikey's got mad skills - Only Mikey that gets any respect! They all have mad skills (they're universally the most gymnastic turtles) but only Rise's make the theme song. It's Leo who just gets a reference to his personality, and a not very flattering one at that.