Should a wife be informed that her husband's care for her was for the sake of Allah the Almighty, or for her sake and his love for her? And should "in Allah" be added when saying "I love you"?

You are not obligated to tell your wife that your care for her was for the sake of Allah, and not for her sake. Your silence regarding her gratitude to you does not affect your righteous intention. Similarly, you are not required to say "I love you for the sake of Allah," but if you say it with the intention of complying with the command, you will be rewarded.