To the precision of the ping arc:
GSM base stations have to know the timing delay between cell phone and base station. This timing delay is determined by "pinging" them. The precision of this measurement gives the distance between phone and base station with an accuracy far better than 550 meters.
That is an example for what precision is possible with everyday consumer electronics. I don't know the exact specs for inmarsat, but taking into account unfavorable geometry, I would expect a precision in the range of better than 5 km.
Given that the satellite is at an altitude of about 36,000 kilometres, a 5km precision would be about +/- 0.014% of the distance, which would be truly remarkable.