Ghibli music has a tendency to be remixed, remixed, and remixed some more... Then sold in countless compilations. There are whole areas in CD shops dedicated to remixed Ghibli compilations.... So I wasn`t even going to bother looking for it. But - I have that specific CD.
The version playing sounds like the AUN J Classic Orchestra version. If you have iTunes you should be able to listen and check.
It is the 4th track.
I can`t be 100% certain because of the background noise and low volume of your recording, but it sounds
very close.