For real plane cubics, allowing all polynomial automorphisms instead of only affine maps yields exactly the same classification, with a complete list of normal forms.
Cayley, On the classification of cubic curves , Transactions of the Cambridge Philosophical Society
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An Automorphic Classification of Real Cubic Curves
For real plane cubics, allowing all polynomial automorphisms instead of only affine maps yields exactly the same classification, with a complete list of normal forms.