Anna,
The determining factor is whether or not the cell center falls within the polygon (mask). It has nothing to do with percentages. You can verify this by converting your resulting raster to points. You won't see any points outside the polygon.
When you perform the extraction I would encourage you to set the Snap Raster in the Environment Settings back to the original raster you were clipping. If not, your output will be shifted from the original pixels.
Internally, we convert the polygon mask to a raster then use that raster to do the extraction. If you are truly concerned about the area (percentage of area), you can convert your polygon to a raster in advance of the extraction. When you do this, make sure you run the Polygon to Raster tool with the Cell Assignment Type parameter set to Maximum Area. Then use the resulting raster as the mask. When we convert your polygon input to raster behind the scenes we are using Polygon to Raster - expect we use the defaults which is Cell Center.
Hope that helps explain how it works.
Regards,
Eric