I strongly recommend keeping to the standard term "abelian group," even though "commutative group" would be more systematic and sensible. The term "abelian group" is universal - I don't know a single mathematician, book, or paper that uses the term "commutative group" - and people comparing what they read here to what they read anywhere else are just going to be confused, and/or are going to confuse third parties when they ask questions.
I strongly recommend keeping to the standard term "abelian group," even though "commutative group" would be more systematic and sensible. The term "abelian group" is universal - I don't know a single mathematician, book, or paper that uses the term "commutative group" - and people comparing what they read here to what they read anywhere else are just going to be confused, and/or are going to confuse third parties when they ask questions.