You missed the point of the parent comment. They were saying that the definition you found, on wikipedia, is better than the definition of "a monoid object in

You missed the point of the parent comment. They were saying that the definition you found, on wikipedia, is better than the definition of "a monoid object in the category of abelian groups" because the definition on wikipedia doesn't require knowing about tensor products etc. You're proving their point. Wikipedia doesn't define it with that language for exactly the reasons the comment pointed out.