I've attached below a picture of the Columbian, Jeffersonian, or Imperial mammoth (in my opinion, all three are the same species), and looking at it, it's easy to see how it's really just "the American elephant." Compared to Elephas or Loxodonta, it's got a more sloped back, a smaller domed back to its head, and larger tusks, but is otherwise, absolutely a member of the same immediate family.
The Stegodons, on the other hand, are slightly different animals; still members of the Elephantidae family, but part of a different subfamily. Today, if we were to see a living Stegodon or other related animal, we'd think it was an elephant with a few odd features, notably extremely long, straight tusks.
Sigh. Once again I regret the accident of history that allowed me to live in North America after more than half of its fascinating megafauna went extinct.