@soaproot No, it's the same thing. Wikipedia just uses multiplicative notation and talks about (X_1)^(m_r) * … * (X_m)^(m_r) where X_1, …, X_r are formal symbols and (m_1, …, m_r) ∈ ℕ^r, instead of (m_1, …, m_r) directly. I just decided to strip away the multiplicative notation to make the post more widely accessible.

At first they leave the conditions »total order« and »the neutral element is minimal« from my original post away, but they add them in »Definition, details and variants« and mention the relation with well-ordering.