The Science of Oral Traditions

All four New Testament Gospels credit Mary Magdalene and others with being first witnesses to Jesus' resurrection. Paul, writing a couple of decades after Mark, says Peter was the first to see him and there were no women there. Did the leaders of the Jerusalem church lie to Paul about who saw the resurrected Jesus first? Or did the gospel authors have some reason to elevate the status of women in the church? I think it's a mistake to think that the options are lying or telling the truth when you're dealing with oral traditions.