In the lines "After my bout with Cecil Jacobs when I committed myself to a policy of cowardice, word got around that Scout Finch wouldn't fight anymore, her daddy wouldn't let her. This was not correct: I wouldn't fight publicly for Atticus, but the family was private ground", what is revealed about Scout's character?
Let's look at the context of this line in order to better understand Scout's characterization.
First of all, Scout wants to fight with Cecil Jacobs because on the playground, he announces that "Scout Finch's daddy defended niggers" (77). Scout defends her father against what she sees as an insult, but isn't sure how to handle the idea behind the insult itself. She follows up with her father, trying to understand why her father is defending Tom Robinson, who explains that he is doing it because it "affects him personally" (78). If Atticus did not follow up on this case, he would no longer respect himself - and this is in spite of the fact that he knows he is going to lose. During their conversation, Atticus asks Scout to "do one thing for me if you will: you just hold your head high and keep those fists down. No matter what anybody says to you, don't you let 'em get your goat. Try fighting with your head for a change" (78).
Accordingly, Scout then chooses to not fight Cecil even though he calls her a coward because "if I fought Cecil, I would let Atticus down. Atticus so rarely asked Jem and me to do something for him, I could take being called a coward for him" (79). However, at Christmas, Scout's resolve is tested by her cousin Francis, whom she despises. Francis calls Scout stupid and insults Atticus in many ways, including calling him "nothin' but a nigger-lover"(85).
This leads us to your line in question. That quotation explains several elements of Scout's character:
- Scout considers not physically fighting cowardice, which indicates that she has matured in an environment where she has felt the need to prove herself to others. She also does not care that fighting is "unladylike", indicating non-conformity.
- People know that Scout is not fighting because Atticus told her not to, which means Scout either told someone or someone cared enough to find out why she wasn't fighting. The fact that this becomes a source of gossip indicates it is a significant shift, so Scout probably fought a lot before this promise.
- Scout doesn't bother correcting people's misconceptions that she doesn't fight at all or that she doesn't fight because Atticus won't "let" her. She doesn't really care what they say or think since she knows the truth.
- Her promise to Atticus extends only so far. Scout adheres to the adage that family is different, and she is willing to defend her father physically against family members, including her cousin.