You should not use CST6CDT
. That is an old-style POSIX time zone. They're only there for backwards compatibility purposes. The correct zone for US Central Time is America/Chicago
. There's a section about POSIX time zones in the timezone tag wiki if you want to learn more.
You'll also need to tell moment exactly what format you are using, and you'll want to tell it that it is being supplied in UTC.
var stampString = 'January 20 2014 01:03';
var m = moment.utc(stampString, "MMMM D YYYY HH:mm");
var convertedTime = m.tz("America/Chicago").format('LLLL');