How does the improved transistor astable multivibrator work?
I can't see what the problem is here... "When TR2 is on causing the right pin of D2 to be grounded", C2 is fully charged ("-" at the left plate and "+" at the right) plate. The diode brings down its right plate and fixes it to +0.7 V. The capacitor begins to recharge and its left plate gradually increases until reaches about +0.6 V when TR1 begins turning on... and the self reinforcing positive feedback begins acting. The circuit rapidly switches... TR2 cuts off ... D2 cuts off as well... and disconnects C2 from the TR2 collector thus improving the rising edge...

So your mistake is in this assertion - "As C2 is not charged yet, the right and left pin of it will be 0.7V." C2 is charged but with the opposite polarity... and its left pin is at -8.3 V.