Most people probably don’t give as much thought to this question as we’re giving it. These are all just terms of convenience, for geologists, geographers, and geography teachers alike. The division between North and South America is usually explained in terms of the Isthmus of Panama, just as Africa is usually set off based on the Isthmus of Suez, but not because there’s a rigorous global standard for continental separation.
Our tagging scheme for places has an inherent bias, by including place=continent but not including any tag for a similarly sized cultural or geopolitical region like Latin America or the Middle East. Arguably these regions matter more than the continents for some purposes. Maybe there was an assumption was that the continents are better defined, and thus a better fit for a data-driven project like OSM. But we’re realizing that the continents are only as well defined as other macrogeographical features like seas and mountain ranges – that is, sometimes but not always.
We don’t have a perfect answer for how to handle such large features, because we don’t have any way to model amorphous features other than as an arbitrary point. We probably have to accept some degree of arbitrariness at this scale, but “splitting the difference” and including both systems of dividing the western hemisphere seems like a decent compromise to me.