I'm not surprised this is confusing.

Usually the subscript "$1-\alpha$" indicates the chi-square value is at the $1-\alpha$ quantile of the distribution.

So that would suggest it has an area $1-\alpha$ to its left and $\alpha$ to its right.

That is to say, your option 1.

But on the other hand "$\alpha$-quantile would more usually suggest the lower tail, your option 2.

I'm not surprised this is confusing.

Usually the subscript "$1-\alpha$" indicates the chi-square value is at the $1-\alpha$ quantile of the distribution.

So that would suggest it has an area $1-\alpha$ to its left and $\alpha$ to its right.

That is to say, your option 1.

But on the other hand "$\alpha$-quantile" would more usually suggest the lower tail, your option 2.

So which is it:

Well, the statistic should reject when correlations are large, and the formula will produce large values of the test statistic when correlations are large.

So when the population correlations are zero, the upper tail of the null distribution, of area $\alpha$ forms the rejection region, so you need $\alpha$ in the upper tail.

Which is your option 1.