I think the confusion here is caused by the various writers, editors, etc who have deliberately shied away from making Wolverine a superhero in the traditional sense. Because, let's face it, a lot of comics are written and drawn today by people who find the concept of superheroes embarassing and juvenile.
With Wolvie, they've tried to have their cake and eat it, too. He wears a costume (usually), he has superpowers and he typically fights characters demonstrably more amoral than himself. On the other hand, he doesn't stop bank robberies, give speeches about virtue and courage, or hide his identity from a snoopy girlfriend. It's a formula that's made him a favorite character for modern readers, who delight in the vicarious thrills of super-goings on but whose reaction to moralizing of any sort ranges from boredom to hostility.
But in the end, Wolverine is a superhero no matter how much Marvel tries to downplay it. If it walks like a duck, quacks like a duck and smokes a cigar like a (Howard the) duck, then it's a duck.
The "fights crime" requirement is one very few if any Marvel "heroes" would meet. You almost have to go back to Lee and Ditko's Spider-Man to find a Marvel character who spent any real time stopping bank heists or muggings. By now, the Marvel heroes seem (to me anyway) to be exclusively devoted to fighting costumed bad guys trying to blow up worlds, screw with time or loose monsters in the streets or whatever. The don't fight "crime" any more than James Bond does; they're world-savers, not lawkeepers. (I assume murders, rapes, assaults and robberies are handled in the Marvelverse the same way they are on our own Earth; by the police.)
I suppose you could change it to "fights evil," though again it gets cloudy. Magneto and his band have a more violent and radical agenda than Xavier, but does that make it inherently evil? (Well it did in the old days, when Maggie had the courtesy to name his group the Brotherhood of EVIL Mutants.
) Marvel goes to some effort to paint the world in shades of gray, to the point I'm not anything qualifies as truly evil. The job of Marvel heroes seems more along the lines of limiting damage to civilians and property, or solving differences of philosophy through fisticuffs. In fact, for the last year they've been exclusively devoted to fighting each other in "Civil War."
I'd argue a majority of comics today don't feature "superheroes" in the traditional sense. I'm ready to adopt the less partisan appelation, "Meta-Humans."
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