I see the situation as being closer to the following: in many cases, people are using "morality" to refer to the same concept, and are disagreeing over the nature of that concept.

When many people agree that murder is wrong but disagree on the reasons why, you can argue that they're referring to the same concept of morality but confused about its nature. But what about less clear-cut statements, like "women should be able to vote"? Many people in the past would've disagreed with that. Would you say they're referring to a different concept of morality?