You are correct there is conflicting information, but I suspect general agreement. My take on why this split, is focus of explanation, that is, I think all your authors primarily agree, but are explaining differing concepts. Most play, whether on offense or defense, favors balanced divisions. This is mainly due to the March to the Sound of Guns feature where each corps ought to be able to hold it own for at least one round of battle prior to the rest of the army arriving. If divisions are relatively equal, then shifting them from corps to corps is relatively simple. But this meaning of divisions is infantry divisions. Pure infantry divisions are not as strong as one with at least one cavalry and some artillery.
Specialized artillery divisions can often stand off from losses, they are very fragile. I do build one for the Army of the Potomac.
Of course cavalry division are a totally separate question. All infantry division need one cavalry to help them 'see' and engage; but all other cavalry are more useful in independent cavalry divisions, so this is where your embedded cavalry contribute to a balanced division. This frees the pure cavalry units for cavalry divisions. Which can function as independent divisions or join for a cavalry corps raid. For pursuit after winning a battle it doesn't matter where cavalry is located in the corps.
So I am saying, infantry divisions need a cavalry and some artillery to be solid; an artillery division is ok; and cavalry divisions are desirable, but not at the expense of any divisions without any cavalry.