What is the correct way for a latecomer to perform prayer: Is it by "completion" (considering what he caught up with as the beginning of his prayer), or by "making up" (considering what he caught up with as the end of his prayer and making up what he missed), or by "making up" in utterances and "completion" in actions?

What a latecomer catches up with the Imam is considered the beginning of his prayer, and what he does after the Imam's salutation is considered its end. This is based on the Prophet's (peace be upon him) saying: "Whatever you catch up with, pray, and whatever you miss, complete." And completing something can only occur after its beginning.