Let $M$ be a c.t.m., $\mathbb{P}\in M$ be a poset, $p\in\mathbb{P}$ be a condition, and $\tau\in M$ be a $\mathbb{P}$-name. We can define a relation "$p$ determines $\tau$" recursively as follows: $p$ determines $\tau$ iff for all $(\sigma,q)\in\tau$, $p\perp q\,\lor\,(p\le q\,\land\,p$ determines $\sigma)$. It is clear that if $p$ determines $\tau$ then $\tau_G$ is the same set for every generic filter $G$ such that $p\in G$. Also, it is possible to define a $\mathbb{P}$-name $\tau$ that is determined by each of two conditions $p$, $q$, but $\tau_G\neq\tau_H$ whenever $G,H$ are generic filters such that $p\in G$ and $q\in H$.
I have two questions.
If $\tau_G$ is the same set for every generic $G$ containing $p$, is it necessary that $p$ determines $\tau$?
If $p$ determines $\tau$ and $G$ is a generic filter such that $p\in G$, does it follow that $\tau_G\in M$?