Stephenie Meyer has just posted the following on her website, effectively denying the Variety tweet's accuracy.
Just wanted to clear up a little rumor making the rounds. I have absolutely no issues with anyone at Lionsgate. Far from being a negative thing, it’s actually a very nice feeling to know that both Lionsgate and Summit would like to make another Twilight movie if I were ever to write another Twilight book. So, thanks for the support, Lionsgate and Summit. I look forward to working for the first time with new friends at Lionsgate and working again with my old friends at Summit!