terminology - p-split Hecke characters

Let $K$ be a quadratic imaginary field, $bf n$ an ideal in the ring of integers
${cal O}_K$ and $xi$ an algebraic Hecke character of type $(A_0)$ for the modulus $bf n$. One knows (from Weil) that there exists a number field $E=E_xisupseteq K$ with the property that $xi$ takes values in $E^times$.

Let $p$ be a prime that splits in $K$. Consider the following condition: there exists an unramified place $vmid p$ in $E$ with residue field $k_v={Bbb F}_p$ such that $xi$ takes values in the group of $v$-units in $E$.

The condition implies the existence of a $p$-adic avatar of $xi$ with values in ${Bbb Z}_p^times$.

I would like to know:

1) to the best of your knowledge, has been this condition considered somewhere? does it have a "name"?

2) I'm tempted to say that $xi$ is $p$-split if the condition is satisfied (and that $v$ splits $xi$). Would this name conflict with other situations that I should be aware of?