Now think if they pull the winning numbers first, ( the difference is instead of only 6 numbers the numbers are thousands) and we need to buy random tickets looking for the right combination of numbers before anybody else does. Your one ticket hits 1 number, not enough to win, but that is your best ticket, now you buy another, and you hit 3 numbers, this is still not enough to win, but this is your new best. Every new block @10 mins is a new lotto drawing, your old best number from another block means nothing toward the new block other than a way to show how close you had come to winning. The lower your hash rate is like only buying one ticket compared to the mega farms/pool buying millions of tickets every 10 minutes or for every new block.
When Apollo shows 13B Best Shares, does it mean that I managed to pull 13B winning tickets, that I managed to FIND a Valid Block at the current network difficulty level BUT did it too late?
You mentioned 12B accepted shares so your Apollo managed to pull 12B loser tickets and 13B is your highest loser ticket.
In a pool this would be your proof of work and would be used to calculate the reward.
For solo mining these values are for information purposes only. Except for rejected shares, too many may indicate an error.