Great question, Yubraj. The dictionary answers that you referenced above are correct but delivered in a way that is rather difficult for even native speakers to understand.

Quite simply, "after all" is used to show that something is true or happened despite reasons to believe otherwise. It's a cue **to tell the reader about an idea/action continuing *despite* something else**.

To address your examples above:

 1. *"The rain has stopped, so the game will go ahead after all."* → We might have expected that the game would be cancelled because of the rain. Instead, it's going to continue.
 2. *"I do like her - after all, she is my sister."* → There may have been a question about whether I like my sister. At the end of the day, however, I must like her because she is my sister.
 3. *"Maybe she was right after all."* → I didn't believe that she was right before, but now I think that I was wrong about her.
 4. *"I'm sorry, but we’ve decided not to come after all."* → We intended to be there, but something changed and we are not going to attend.

Now there's another usage which the Macmillan Dictionary captured in the two examples below. In this usage, the intention is **to emphasize a point by adding an additional reason.**

 5. *"She shouldn’t be working so hard – she is 70, after all."* → She shouldn't be working so hard, especially considering that she is so old.
 6. *"I’m not really ambitious. After all, money isn’t everything."* → I don't need to be ambitious, especially when there are other priorities in life. (This writer is equating ambition with money.)

I hope that this helps.