IMHO the answer to “where polynomials are initial” (not to the title question, which is too broad for me) is already given in “Awodey. Category theory. 9. Adjoints. 9.3. Examples of adjoints. Example 9.10.”

In that example, the adjunction of functors is constructed, where its free (left adjoint) functor $F$ goes from the category of rings (=RingCat) to the category of rings with distinguished element (= pointed rings). If we define this adjunction via unit ($\eta$), then the definition says that

for every object $R$ in RingCat (= for every ring $R$) there is an initial object in the category $select(R)\downarrow U$ (comma category),

where $U$ is the forgetful functor.

Furthermore, a chosen initial object for $R$ consists of $F(R)$ (= $R[x]$) and $\eta(R):R\to U(F(R))$ (= a ring homomorphism constructing constant polynomials). The distinguished element in $F(R)$ is the identity polynomial.

IMHO the answer to “where polynomials are initial” (not to the title question, which is too broad for me) is already given in “Awodey. Category theory. 9. Adjoints. 9.3. Examples of adjoints. Example 9.10.”

In that example, the adjunction of functors is constructed, where its free (left adjoint) functor $F$ goes from the category of rings (=RingCat) to the category of rings with distinguished element (= pointed rings). If we define this adjunction via unit ($\eta$), then the definition says that

for every object $R$ in RingCat (= for every ring $R$) there is an initial object in the category $select(R)\downarrow U$ (comma category),

where $U$ is the forgetful functor.

Furthermore, a chosen initial object for $R$ consists of $F(R)$ (= $R[x]$) and $\eta(R):R\to U(F(R))$ (= a ring homomorphism constructing constant polynomials). The distinguished element in $F(R)$ is “$x$” (the projection polynomial).