I was discussing ASOIAF and the following statement came up: ‘He (Twyin) doesn’t care what others think.’ Do you think this statement is accurate?

Absolutely not. Due to his childhood growing up in Tytos’ court, Tywin is (if anything) hyper-sensitive to the opinions of others:

“Tywin mistrusted laughter. He heard too many people laughing at your grandsire.“

Most of Tywin’s life is about stamping an idea of House Lannister into the minds of others: hence not only carrying out the Rains of Castamere but making sure a song was sung and sung widely, hence launching all-out war on the Riverlands because Tyrion was arrested, and on and on, trying to silence that laughter.