It would have been helpful if you would have included the different parameters specified in the project raster and resample tools. Judging from the effect and you data (0, 1?) you are using the nearest resampling technique (which is OK for this type of data). The original data is visualized using a on-the-fly projection (skewed pixels). I also see that the area has been moved south about 6 pixels, did you move the area, or is this what happened after projection (that would be something to check). Is there a specific transformation you apply?
Another question: what does the data represent? The reason I ask is that if the dataset is not too big and it originally represents points or areas, you could consider converting back to vector format, project and then rasterize. Or even much better, if you have access to the original vector input data you should use this instead (project and then convert to raster)
One of the differences you could see, is when you first resample and then project. This will create a different result. Projecting the raster first, yields "comparable" single pixels. If you resample afterwards (factor 0.5) there will be 4 output pixels for each input pixel, but graphically you don't see the difference.
Doing this in a single step applies a different order. It is projected and resampled in memory and then the output pixels are assigned based on the center of input pixels.