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What made me think that these definitions are equivalent is that Whitherspoon invoke in the proof of Theorem 3.2.6 (HKR Theorem) a result from Weibel's book on smooth algebras, although she gave a different definition for this notion.
No, for me a $k$-algebra $A$ is only a $k$-vector space endowed with a bilinear map $A \times A \rightarrow A$. I should have mentioned it in the question.
