Typing relations terminology – how do I read typing relations?
The two inference rules are different, because the first requires that x:T_1 is the only assumption, while the second allows side assumptions.  This can have subtle effects of the consequence relation for the type theory prevents the type theory from modelling weakening by having as the hypothesis rule:

$$
\frac{}{\Gamma, x:A \vdash x:A}
$$

In your English gloss, I would have under the sole assumption, to avoid confusion with the assumption-gobbling inference rule:

$$
\frac{\Gamma, x:T_1 \vdash t_2 : T_2}{\vdash \lambda x:T_1.t_2:T_1\to T_2}
$$