Equivalence of Definitions of Polynomial Ring in One Variable

Theorem
Let $R$ be a commutative ring with unity.

The following definitions of polynomial ring are equivalent in the following sense:
 * For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates.

Outline of Proof
We show that they all satisfy the same universal property.

Proof
Use Polynomial Ring of Sequences Satisfies Universal Property

Also see

 * Rings of Polynomial Forms Isomorphic