I know that this post is a little long and I beg your indulgence on account of that but it is worth reading, I think, because it is the view of Albert Barnes who was a Presbyterian minister and able commentator. He offers this explanation for 2 Cor 5:20
2 Corinthians 5:20
Now then we are ambassadors for Christ - We are the ambassadors whom Christ has sent forth to negotiate with people in regard to their reconciliation to God, Tyndale renders this: Now then are we messengers in the room of Christ. The word used here πρεσβεύομεν presbeuomen, from πρέσβυς presbus an aged man, an elder, and then an ambassador) means to act as an ambassador, or sometimes merely to deliver a message for another, without being empowered to do any thing more than to explain or enforce it - Bloomfield. See Thucydides 7, 9. An ambassador is a minister of the highest rank, employed by one prince or state at the court of another, to manage the concerns of his own prince or state, and representing the dignity and power of his sovereign - Webster. He is sent to do what the sovereign would himself do were he present. They are sent to make known the will of the sovereign, and to negotiate matters of commerce, of war, or of peace, and in general everything affecting the interests of the sovereign among the people to whom they are sent.
At all times, and in all countries, an ambassador is a sacred character, and his person is regarded as inviolable. He is bound implicitly to obey the instructions of his sovereign, and as far as possible to do only what the sovereign would do were he himself present. Ministers are ambassadors for Christ, as they are sent to do what he would do were he personally present. They are to make known, and to explain, and enforce the terms on which God is willing to be reconciled to people. They are not to negotiate on any new terms, nor to change those which God has proposed, nor to follow their own plans or devices, but they are simply to urge, explain, state, and enforce the terms on which God is willing to be reconciled. Of course they are to seek the honour of the sovereign who has sent them forth, and to seek to do only his will. They go not to promote their own welfare; not to seek honour, dignity, or emolument; but they go to transact the business which the Son of God would engage in were he again personally on the earth. It follows that their office is one of great dignity, and great responsibility, and that respect should be showed them as the ambassadors of the King of kings.
As though God did beseech you by us - Our message is to be regarded as the message of God. It is God who speaks. What we say to you is said in his name and on his authority, and should be received with the respect which is due to a message directly from God. The gospel message is God speaking to people through the ministry, and entreating them to be reconciled. This invests the message which the ministers of religion bear with infinite dignity and solemnity; and it makes it a fearful and awful thing to reject it.
We pray you in Christs stead - (υπερ χριστου huper Christou). In the place of Christ; or doing what he did when on earth, and what he would do were he where we are.
Be ye reconciled to God - This is the sum and burden of the message which the ministers of the gospel bear to their fellow-men; see the note on 2Cor 5:19. It implies that man has something to do in this work. He is to be reconciled to God. He is to give up his opposition. He is to submit to the terms of mercy. All the change in the case is to be in him, for God cannot change. God has removed all the obstacles to reconciliation which existed on his part. He has done all that he will do, all that needed to be done, in order to render reconciliation easy as possible. And now it remains that man should lay aside his hostility, abandon his sins, embrace the terms of mercy, and become in fact reconciled to God. And the great object of the ministers of reconciliation is to urge this duty on their fellow-men. They are to do it in the name of Christ. They are to do it as if Christ were himself present, and were himself urging the message. They are to use the arguments which he would use; evince the zeal which he would show; and present the motives which he would present to induce a dying world to become in fact reconciled to God.
