A function $f:\mathbb{N}^k\rightarrow\mathbb{N}$ is arithmetic iff its graph is arithmetic, i.e., there is a formula $\psi(\vec{x},y)$ in the language of Peano arithmetic such that for all $\vec{a}$ in $\mathbb{N}^k$, $f(\vec{a})=b$ iff $\mathbb{N}\models\psi(\vec{a},b)$.
For $f$ to be represented in PA, we have the (much) stronger condition that for all $\vec{a}$ in $\mathbb{N}^k$, $f(\vec{a})=b$ iff $\text{PA}\vdash\psi(\vec{a},b)$.
I'm wondering about an intermediate condition: $f$ is arithmetic, and in addition, PA proves that the associated formula defines a function. That is, $\text{PA}\vdash\forall\vec{x}\,\exists!y\,\psi(\vec{x},y)$.
Would it be reasonable to call this condition expressible in PA? Or would that conflict with generally accepted terminology?
I'm also assuming that definable in PA is a synonym for arithmetic. But I'm interested in hearing from anyone who doesn't agree with this.
Your condition is called definable. This is standard usage in model theory: Given a theory $T$, a definable function is a formula $\varphi(\overline{x},y)$ such that $T\vdash \forall \overline{x}\, \exists^! y\, \varphi(\overline{x},y)$, up to provable equivalence in $T$. Then for any model $M\models T$, $\{(\overline{a},b)\in M^{n+1}\mid M\models \varphi(\overline{a},b)\}$ is the graph of a function $M^n\to M$ (which we abuse terminology by also calling a definable function).
It would be a big mistake to make "definable in PA" a synonym for "arithmetic". The point is that the definition of arithmetic has nothing to do with the theory PA, it's just about the standard model $\mathbb{N}$. On the other hand, the definition of definable has nothing to do with the standard model $\mathbb{N}$, it's just about the theory PA. These are totally different concepts.