Does Tywin really care about the Lannisters’ reputation? On one hand he does everything to make the Lannister name feared and respected yet, on the other he does commit horrible atrocities to taint the Lannister name.
He doesn’t see it as tainting the Lannister name. Tywin believes in the power of fear.
King’s Landing had never loved Lord Tywin. He never wanted love, though. “You cannot eat love, nor buy a horse with it, nor warm your halls on a cold night,” she heard him tell Jaime once, when her brother had been no older than Tommen.
- Cersei II, AFFC
So improving Lannister reputation through the love of his subjects is out. Tywin’s no Ned Stark, nor even a Robert Baratheon in this respect.
More than anything, though, Tywin hates the idea of being laughed at. He considers laughter the death of respect; he saw how his father’s bannermen mocked him and how weak a ruler his father was. As Kevan describes the situation to Tyrion,
“Our own father was gentle and amiable, but so weak his bannermen mocked him in their cups. Some saw fit to defy him openly. Other lords borrowed our gold and never troubled to repay it. At court they japed of toothless lions.”
- Tyrion IX, ASoS
The end result is a man who went way too far in the other direction. Consider Tywin’s introduction.
Tywin Lannister, Lord of Casterly Rock and Warden of the West, was in his middle fifties, yet hard as a man of twenty. Even seated, he was tall, with long legs, broad shoulders, a flat stomach. His thin arms were corded with muscle. When his once-thick golden hair had begun to recede, he had commanded his barber to shave his head; Lord Tywin did not believe in half measures.
- Tyrion VII, AGoT
It takes but four sentences to see that Tywin’s so afraid of mockery that when his hairline recedes, as male hairlines are often wont to do, he’d rather shave his head than have people snigger at him going bald. Two sentences later, this is backed up again in Tyrion’s narration:
A fool more foolish than most had once jested that even Lord Tywin’s shit was flecked with gold. Some said the man was still alive, deep in the bowels of Casterly Rock.
Old Merryweather had been nattering about raising the duty on wine when Lord Rykker said, “If we need gold, His Grace should sit Lord Tywin on his chamber pot.” Aerys and his lickspittles laughed loudly, whilst Father stared at Rykker over his wine cup. Long after the merriment had died that gaze had lingered. Rykker turned away, turned back, met Father’s eyes, then ignored them, drank a tankard of ale, and stalked off red-faced, defeated by a pair of unflinching eyes.
- Cersei II, AFFC
He really can’t stand it, and his reaction is dramatic to the point of absurd. Likewise, he cannot stand the idea of people ever seeing him as weak.
Tywin’s idea of strength, however, is a very narrow, rigid one that involves a hell of a lot of violence. Bannermen making trouble? Collapse their castle, flood their mines, kill them all, promote the song that tells everyone about it. Et cetera, et cetera, all the way up to AGoT. When Catelyn arrests Tyrion he can’t possibly insist the Starks bring the matter before Robert, he has to sack the Inn at the Crossroads and start murdering civilians in the Riverlands. Then when war breaks out he can’t just fight, he has to do this:
“Let them,” Lord Tywin said. “Unleash Ser Gregor and send him before us with his reavers. Send forth Vargo Hoat and his freeriders as well, and Ser Amory Lorch. Each is to have three hundred horse. Tell them I want to see the riverlands afire from the Gods Eye to the Red Fork.”
- Tyrion IX, AGoT
Otherwise he’d be a laughingstock. Right? Right.
Lord Tywin’s fingers laced together under his chin. “Ser Gregor has served us well. No other knight in the realm inspires such terror in our enemies.”
- Tyrion VI, ASoS
This line is said in a discussion between Tyrion and Tywin after news of the Red Wedding breaks in King’s Landing. Note that Tywin considers the terror Gregor inspires to be serving House Lannister.
So yes, Tywin cares deeply about the reputation of House Lannister. He cares about the reputation of House Lannister more than he cares about the members of House Lannister.
“We all dream of things we cannot have. Tywin dreamed that his son would be a great knight, that his daughter would be a queen. He dreamed they would be so strong and brave and beautiful that no one would ever laugh at them.”
- Jaime VII, AFFC
He just doesn’t see that the brutality he used to make House Lannister feared has also deeply hurt each of his children (thereby sabotaging all his dreams for them) and made the Lannisters loathed and mistrusted.