Suppose you have a proof along the lines of
$$\begin{array} {rc} \text{Assume:} & x > 2 \\ & \vdots \\ & \text{Some logic stuff} \\ & \vdots \\ \text{Conclude:} & x > 1 \\ \end{array}$$
Two common ways for this to be interpreted are (1) $(\forall x~.~x > 2) \to (\forall x~.~x > 1)$ and (2) $\forall x~.~(x > 2 \to x > 1)$. Logics that intend the first way include PRA and Hilbert style FOL. Logics that intend the second way include Fitch style Natural Deduction.
Is there common terminology to distinguish the two approaches to interpreting how the free variables are shared between propositions in a proof? Or if you have reason to think there are no such common terms, what would you suggest?