It should be "Hedges' g" rather than vice versa. Personally, I find the original sentence a bit misleading and would re-write it (for a language I knew) as e.g. "Because the sample sizes were different, the corrected effect size (Hedges' g) was calculated using the weighted pooled variance"
In fact, there's rarely any need to explain a standard usage, the formula is self-explanatory!
The subscript version is fairly rare in the literature (but is perfectly acceptable) - there's an example at the end of Section 2 of
http://www.psychometrica.de/effect_size.html Whether to leave it or change it really depends on the context - if it's in a formula I'd leave it. But if it's in text I'd change it.