What is the correct method for calculating prayer times in Western countries, given the discrepancies among available calculation methods and the difficulty for non-specialists to understand astronomical observatories?

The times for prayers are determined by the Shar'i cosmic signs. If you are unable to ascertain them, refer to the nearest reliable Islamic center in your country. If none exists, then rely on calendars issued by trustworthy sources such as the Muslim World League, Al-Azhar, or the University of Karachi. Their differences do not matter unless it becomes clear that one of them contradicts the Shar'i signs. For Allah does not burden a soul beyond its capacity.