When the FAA said that pilots require 20/20 vision, does that mean that you have to achieve 20/20 vision with glasses or without glasses?
It can be with correction. And note that for third class, distant vision only has to be correctable to 20/40. The near vision standard is 20/40 for all classes. See https://www.faa.gov/about/office_org/headquarters_offices/avs/offices/aam/ame/guide/standards/
As @Pondlife notes, the operative words here are, "with or without correction."