Your target is to describe the binned data by providing a median value.
Do you actually want to describe the binned data or the original data? Is there a special reason for the binning?
I imagine that in most situations binning is just a means to an end, e.g., to simplify some other part of your analysis or when using algorithms that only accept a small number of unique values per variable. In that case, you essentially sacrifice information for simplicity.
However, you don't need to bin your data to provide the median of your dataset. You already aggregate your data to a single statistic by computing the median. There is no need to do another aggregation step beforehand, it would just decrease the accuracy of your median. Therefore, you should report the median of the unbinned data.
There might be situations where you would want to report the median of the binned medians: in a hypothetical situation where you collected two datasets of two populations, one where data is already binned and you don't have the original data and one where data is unbinned. If you want to compare the medians of the two datasets, you could consider first binning the yet unbinned dataset and then compute the median. However, even here it would be just throwing away information about one of the populations.
In conclusion, I would report the median of the unbinned data.