What is the correct ruling for Istinja, and must I re-perform past prayers that I offered in a way I thought was correct for removing the traces of impurity?

If you pour water over the spot of impurity so that it covers it, then this is what is obligatory. Wiping alone is not sufficient. What is obligatory is to wash the place by pouring water until its roughness returns. The predominance of belief is sufficient. If the washing water flows over the impure spot, then the istinja’ (cleansing) is correct. Otherwise, you must perform correct istinja’ in the future. As for your missed prayers, you are not required to make them up, according to Shaykh al-Islam Ibn Taymiyyah, for someone who unknowingly omitted a condition.