If you look closely at your picture of the logo, you see that the blue turns to black at the edges - same goes for the orange colour. The computer program you're using does not bring this back to the three basic colours, so the contrast doesn't show right.
You could either find another (more stylistic/basic) logo, or clean up the chart that is given. For example: read the different shades of blue all as being navy. Take stitch 14, 19, 24 and 29 as white. I am not sure how to go about the letter C though... I imagine you'd want the little thorn in the left to show as it is quite essential for the logo...
I think I myself would just take a raster and try to get it right by colouring it myself.
Maybe some-one on this forum has a pattern for you? It looks quite standard... I am sorry not to be of more help.