Is my feeling of disdain towards the words of my relative, who asked for the Quran to be turned off and music to be played, considered self-admiration (ʿujb) that invalidates my deeds, and is seeking forgiveness for her sufficient, or must I ask for her forgiveness?

What the woman said was an evil and objectionable statement. You should have advised her gently and kindly, explaining that happiness lies in intimacy with Allah, and that listening to music and musical instruments is forbidden. Forbidden music is an illusory, temporary pleasure followed by regret and sorrow, especially if it diverts a servant from the remembrance of Allah. "And whoever turns away from My remembrance – indeed, he will have a difficult life, and We will gather him on the Day of Resurrection blind." (Taha: 124). Your strong disapproval of what she said is a good feeling, indicating the vitality of your heart and your veneration of the rituals of Allah. "That [is so]. And whoever honors the symbols of Allah – indeed, it is from the piety of hearts." (Al-Hajj: 32). It is not necessary for this to be blameworthy arrogance, as long as it stems from zeal for the Book of Allah. It is not clear that this necessarily involved taunting her, but if you said it to taunt her, then seek Allah's forgiveness. You do not owe her anything that requires you to ask her for forgiveness. For intangible rights, seeking forgiveness for the wronged party is sufficient. Seeking forgiveness for her is a good deed in any case.