I think that despite the story talking constantly about "honesty" and "lies", words like "selflessness" and "greed" would be more appropriate.
More than honest, Nao is selfless. Before Liar Game, she was honest because she lacks the greed that would make lying useful to her.
Now that she's in the game, she's learning to lie. But while the immediate beneficiaries of those lies are her, her group and her ideals, the ultimate goal of those lies is to arrive at a good outcome to as many people as possible, even at her own expense. And this is what Akiyama is talking about.
The reason why the others can trust her is because even if she does trick or betray them during the game (or tacitly agree with Akiyama underhanded ways), as long as she win big enough, she WILL fulfill the ultimate promise of helping them get out of the Liar Game debt-free. This is where she absolutely won't entrap or betray them.
And regarding getting back at others, I'd say it's in relation to the "ultimate getting back" which would be bankrupting the person. She does want take Yokoya down a few pegs, make him lose the game, maybe even make him lose hard, but she doesn't want to get back at him by ruining his life. She knows he can afford to lose that money and she would return it after the game if he couldn't.
Same with other characters. She will prioritise saving her friends, those in her plans and those that are nice to her, but she will also forgive and save the ones that were terrible to her if she has enough money for it.