I was under the impression that the Tabulate Area tool was for use with raster data but I now see it can do both.
I took a closer a look and when working with vector data, both the Tabulate Area and Summarise Within tools rasterize the data and a cell size needs to be set for Tabulate Areas.
I checked these three tools against my own data and the difference was minor, around 0.007%. Tabulate Areas was less similar than the other two but I suspect if I had to decrease cell size, it would match up.
I then checked these three tools on the dataset you provided on Mangroves in central Mozambique and again the results were consistent.
For example:
Tabulate Intersection: 85 376 ha of mangroves
Tabulate Areas (50 ha cell size given fragmented extent of mangroves): 85 333 ha (I converted to ha)
Summarise Within: 85 376 ha
Therefore results from Tabulate Intersection and Summarise Within are identical and Tabulate Areas only 0.05% off.
Perhaps try again in a new view?