SharePoint site, any kind of file (tried xlsx & txt) → can edit only from Internet Explorer (11). Chrome, Firefox, and Opera allow only download. See bottom left option difference in attached screen shots.
OpenDocumentsReadWriteWhileBrowsing to registry (per some blogs/forums) didn't help.
NOTE: I think I've encountered other SharePoint sites where any browser allows editing a file. Will update post if I find that to be true.