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Tywin actually disowned Jaime there. He wanted Jaime to do his duty to his family, Jaime refused. Tywin noted that he has tried for forty years to teach him and if Jaime has not learnt by now, he never will. Kingsguard only forsake their claims, not their family names like Maesters do. His vows don't make him any less part of House Lannister. Its his insistence that his duty as KG comes above his duty as a Lannister that made Tywin disown him.
Tywin: You can't serve in the Kingsguard with one hand.
Jaime: Where's that written? I can and I will. The Kingsguard oath is
for life.
Tywin: The war is over. The king is safe.
Jaime: The king is never safe. How many people in this city alone
would love to see his head on a pike?
Tywin: Other knights protected the king while you were a prisoner.
They will continue to do so when you go home.
Jaime: Home?
Tywin: You'll return to Casterly Rock and rule in my stead.
Jaime: You are the Lord of Casterly Rock.
Tywin: I am the King's Hand. My place is here. I don't expect to see
the Rock again before I die.
Jaime: You know what they call me? Kingslayer. Oathbreaker. Man
without honor. Now you want me to break another sacred vow.
Tywin: You won't be breaking anything. There is a precedent to relieve
a Kingsguard of his duties. The king will exercise that prerogative.
Jaime: No.
Tywin: No?
Jaime: No.
Tywin: I don't believe I asked you a question.
Jaime: There's my answer.
Tywin: If you think your bloody honor comes before--
Jaime: My bloody honor is beyond repair, but my answer is still no.
I don't want Casterly Rock. I don't want a wife. I don't want children.
Tywin: What do you want?
Jaime: Supper would be nice.
Tywin: For 40 years I've tried to teach you. If you haven't learned
by now, you never will. Go. If serving as a glorified bodyguard is the
sum of your ambition, go serve.
Jaime: I suppose you want the sword back.
Tywin: Keep it. A one-handed man with no family needs all the help
he can get.
Season 4 - Two Swords
Jaime took the hint and said as much to Cersei. Cersei disagreed however.
Cersei: And now I'm marrying my eldest son to a wicked little bitch
from Highgarden while I'm supposed to marry her brother, a renowned
pillow biter.
Jaime: So Father disowned me today.
Cersei: He can't disown you. You're all he's got.
Season 4 - Two Swords
Turned out Cersei was right. Tywin accepted Jaime's offer to become heir to Casterly Rock if he let Tyrion live afterwards.
Books
This is how it happened in the books, it was more clear that Tywin had disowned Jaime after he refused to do service for his house and when Tywin said he was his son, Jaime said he was a Knight of Kingsguard. So one can say, it was Jaime who burnt the bridges.
“You cannot serve in the Kingsguard without a sword hand -”
“I can,” he interrupted. “And I will. There’s precedent. I’ll look
in the White Book and find it, if you like. Crippled or whole, a
knight of the Kingsguard serves for life.”
“Cersei ended that when she replaced Ser Barristan on grounds of
age. A suitable gift to the Faith will persuade the High Septon to
release you from your vows. Your sister was foolish to dismiss Selmy,
admittedly, but now that she has opened the gates -”
“- someone needs to close them again.” Jaime stood. “I am tired of
having highborn women kicking pails of shit at me, Father. No one ever
asked me if I wanted to be Lord Commander of the Kingsguard, but it
seems I am. I have a duty -”
“You do.” Lord Tywin rose as well. “A duty to House Lannister.
You are the heir to Casterly Rock. That is where you should be.
Tommen should accompany you, as your ward and squire. The Rock is
where he’ll learn to be a Lannister, and I want him away from his
mother. I mean to find a new husband for Cersei. Oberyn Martell
perhaps, once I convince Lord Tyrell that the match does not threaten
Highgarden. And it is past time you were wed. The Tyrells are now
insisting that Margaery be wed to Tommen, but if I were to offer you
instead -”
“NO!” Jaime had heard all that he could stand. No, more than he could
stand. He was sick of it, sick of lords and lies, sick of his father,
his sister, sick of the whole bloody business. “No. No. No. No. No.
How many times must I say no before you’ll hear it? Oberyn Martell?
The man’s infamous, and not just for poisoning his sword. He has more
bastards than Robert, and beds with boys as well. And if you think for
one misbegotten moment that I would wed Joffrey’s widow...”
“Lord Tyrell swears the girl’s still maiden.”
“She can die a maiden as far as I’m concerned. I don’t want her, and I
don’t want your Rock!”
“You are my son -”
“I am a knight of the Kingsguard. The Lord Commander of the
Kingsguard! And that’s all I mean to be!”
Firelight gleamed golden in the stiff whiskers that framed Lord
Tywin’s face. A vein pulsed in his neck, but he did not speak. And did
not speak. And did not speak.
The strained silence went on until it was more than Jaime could
endure.
“Father...” he began.
“You are not my son.” Lord Tywin turned his face
away. “You say you are the Lord Commander of the Kingsguard, and only
that. Very well, ser. Go do your duty.”
ASOS - Jaime VIII
