Polynomial Interpolation Uniqueness

Question

In a problem on my linear algebra homework this week, we were asked to prove that for distinct $x_{1},\ldots,x_{n}\in F$ and $c_{1},\ldots,c_{n}\in F$ (possibly not distinct) for an arbitrary field $F$, there is a unique polynomial function $P:F\to F$ such that $P(x_{i})=c_{i}$ for all $i\leq n$. In my proof (as well as others' proofs), I showed that the coefficients of the polynomial function are uniquely determined, but this I believe shows that the polynomial itself (as in $P\in F[x]$) is uniquely determined, not the polynomial function. Because in finite fields, we can have polynomial functions that agree at all points but whose polynomials don't agree. So in polynomial interpolation, is it the polynomial or polynomial function that is uniquely determined?