They know the pings came from MH370 because each ping has an identifying code unique to that aircraft only.
Furthermore, it was I that mentioned the + or - 300kms and only because I do not know whether Inmarsat use Atomic Clocks on their Satellites and I reckon it is a pretty safe bet that they do, that all satellites do, in which case that error margin goes out the window.
Then that leaves the errors attributed to the calculations in reconstructing the flight which the Aircraft Performance Analysts will be aware of since their models are based on a range of assumptions which are unverifiable but nonetheless there is a method in the science which means none of it can be discounted. They are indeed very lucky that they have the Inmarsat data which gives them 7 start points. If they only had the last primary return then they practically can only draw a circle around that, the radius of which can
only be the best possible range of the aircraft for the given amount of fuel on board.
These methods are not tried or tested but the good people and experts at Boeing, Inmarsat and Rolls Royce are no fools either. Very difficult explaining their methodology to the layman but one day they will reveal the entire intriguing process.