When is “Twilight”?

In Exodus 12:6, the Israelites were commanded to kill the Passover lamb at “twilight.” This phrase has been at the center of a debate that has caused some to observe the Passover on the wrong day – at least according to what we teach in the Living Church of God. 
Why do we observe the Passover in the evening at the beginning of Abib 14, while today’s Jews observe it at the beginning of Abib 15?

Course Spotlight From God’s Feast Days: Passover