Violating Commandments is commonly linked to sin
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.
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
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.
Pay respect where respect is due
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).