How to convert this trace into GPX (correct timestamps) for OSM using gpsbabel?

This seems to be already in GPX format, in principle. Does the upload to OSM not work? I guess it may be due to the timestamps not being in ISO 8601 date-time format (they are e.g. "Sat Jan 18 2014 14:01:32 GMT+0200 (EET)"). If possible you should configure your app to record the time stamps in the correct format, tell the app author or use another one.

I do not know how to use gpsbabel to convert the time stamps. However, I see two alternative options:

1. As an ugly workaround you could replace the timestamps in wrong format by new ones in the correct format. On upload please select the "private" privacy option which hides the time/speed info to other users as it is faked.
   1. You can use gbsbabel to add fake timestamps (see also 1 and 2):
      gpsbabel -i gpx -f gps_logmee.gpx -x track,faketime=f19700101000000+1 -o gpx -F outputfilename.gpx
   2. You could use a regexp-capable text editor and search
      <time>[^<]*</time>
      and replace all by
      <time>1970-01-01T00:00:00Z</time>
2. Alternatively you could use the gpx file directly (without uploading) in most editors (iD, JOSM, Potlatch2). That works even without correct time stamps. Of course that "no upload method" looses the benefit to other mappers.