hairhooks according to the extract of earnings you do not qualify for Silver based on the number of sectors flown on QR (2) BUT you qualify by earning 40 Qpoints on QR where only 30 are required. So with 151 Qpoints earned you qualified for Silver 29 June. Congratulations!
To me it is obvious that when you were told that you had only earned 146 points the person you spoke to was using the registration date, or even the days date, and not the flight date. Try again asking for a retro calculation for 30 June and if you get the same incorrect information ask to speak t0 a supervisor.
sometimes QR register OW partner flights as the flight occurring on the registration date not flight date.
If you look at the flights and posting dates for your QR flights they are the same. But for your BA flights the registration date is much later than the flight date. Sometimes the flight date is not entered and thus defaults to the registration date. A very human error given how QR registers its own flights!