Is this CI $p_1-p_2\in\Big[\hat p_1-\hat p_2\pm z_{1-\alpha/2}\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}}\Big]$
equivalent to
this CI $p_1-p_2\in\Big[\hat p_1-\hat p_2\pm z_{\alpha/2}\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}}\Big]$
?
And the only difference between them is the way of calculating values from the normal table, am I right?
Yes, it is the same as $$z_{1-\frac{\alpha}{2}} = - z_{\frac{\alpha}{2}}$$