Relativization of a name in forcing

Question

What is the difference between

$$\tau \text { is a }\mathbb{P}\text{-name and }\tau\in M$$ and $$(\tau\text{ is a } \mathbb{P}\text{-name})^M?$$ Here $M$ is a ground model. In particular I would like to know which is a stronger notion.

Answer 1

Both are equivalent (in this case). The property of being a name is absolute for transitive models, since it only looks at what's in the transitive closure of $\tau.$ Thus the names according to $M$ are exactly the names that are in $M$.