Is what is said to oneself during prayer, by way of self-encouragement or reminding oneself of a ruling—such as "You are strong; you will overcome the whispers of Shaytan" or "I said I am not afflicted with whispers of Shaytan"—considered a form of self-admiration?

This is not surprising; rather, it is something that ought to be done. Whenever a whisper (waswas) occurs, repel it and convince yourself that you are not a victim of whispers and that you are able to overcome it with the help of Allah. Trust in Allah and remember that the strength to repel whispers and achieve healing comes only from Him.