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For an official teaching, the Catholic Church's teaching is that careless references to God violates the Second Commandment (Third Commandment in some other denominations). Other denominations may have similar points of view, but to keep this answer concise I confine it to the Catholic view.
Exodus 20:7
You shall not take the name of the LORD, your God, in vain. For the LORD will not leave unpunished him who takes his name in vain.
From the Catechism ...
CCC 2143 "The Lord's name is holy." For this reason man must not abuse it. He must keep it in mind in silent, loving adoration. He will not introduce it into his own speech except to bless, praise, and glorify it.
See also Psalm 29:2 (KJV)
Give unto the Lord the glory due unto his name; worship the Lord in the beauty of holiness.
And Psalm 113:2-3 (KJV)
Blessed be the name of the Lord from this time forth and for evermore. From the rising of the sun unto the going down of the same the Lord's name is to be praised.
To use "For God's sake" per your definition 3 -- as a sign of annoyance, impatience, or urgency -- does not bless, praise or glorify God. That usage also goes against the principle of Christian Charity1.
CCC 2146 The second commandment forbids the abuse of God's name, i.e., every improper use of the names of God, Jesus Christ, but also of the Virgin Mary and all the saints.
In closing, St Augustine lays out the respect due to the Lord
[God's] name is great when spoken with respect for the greatness of his majesty. God's name is holy when said with veneration and fear of offending him. ~ St. Augustine, De serm. Dom. in monte 2, 5, 19: PL 34, 1278.
1 388. What is charity? From CCC 1822-1829, and 1844)
Charity is the theological virtue by which we love God above all things and our neighbor as ourselves for the love of God. Jesus makes charity the new commandment, the fullness of the law. “It is the bond of perfection” (Colossians 3:14) and the foundation of the other virtues to which it gives life, inspiration, and order. Without charity “I am nothing” and “I gain nothing” (1 Corinthians 13:1-3).
Upvote:-2
Since it is God, Himself, who enables us 'to do,' (or exist at all for that matter), it would be impossible for us to do anything for His "sake," - i.e. to "help," God. For any person or "thing," to think it could "help," God is an insult ("blasphemy").
Philippians 2:13 - For God is working in you, giving you the desire and the power to do what pleases Him.
John 15:5 - I am the vine; you are the branches. If you remain in me and I in you, you will bear much fruit; apart from me you can do nothing.
Colossians 1:16 - For in Him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things have been created through Him and for Him.