I don't know if this was an invitation for further discussion, but I'm gonna jump in to expand on something, specifically this...
So, it's not that I necessarily disagree with this specific point but I personally can't read Inej's behavior towards Kaz as her assuming she has to earn his approval or appreciation (though of course she would like them) since she's very aware of her own worth as well as of his cruelty, even if she can't pinpoint why he is the way he is. Like, if anything she's upset that he doesn't acknowledge everything she already does with and for him (rightfully so I'd say).
Maybe I'm misinterpreting which specific aspect of their dynamic you're talking about here, but IF we're talking about her trying to find and/or push redeeming qualities in Kaz (according to his POVs) then I'd like to put forward another interpretation of what's going on with Inej's... "insistence" in said regard, at least prior to the Ice Court.
It has to do with this passage from Crooked Kingdom:
In her early days at the Menagerie, she'd believed someone would rescue her. Her family would find her. An officer of the law. A hero from one of the stories her mother used to tell. Men had come, but not to set her free, and eventually her hope had withered like leaves beneath a too-bright sun, replaced by a bitter bud of resignation.
Kaz had rescued her from her hopelessness, and their lives had been a series of rescues ever since, a string of debts that they never tallied as they saved each other again and again. [...]
As established, Suli people rely almost entirely on community and Ketterdam is all about self-interest even within gangs and families; a user on TikTok (@/winterbrekker) made the brilliant observation that Inej and Kaz's exchange in the bathroom scene "Was there anyone to protect you?" / "Was there no one to protect you?" shows how Inej sees protection as something that people should have by principle while Kaz sees its absence as normal.
I did edit this bit in the original post Heleen's emotional abuse very much hinges on this desperate need for a safe place which, among other things, feels like another layer of mockery of Inej's identity and personhood, so while I don't think she has trust issues per se, what she had to endure at the hands of both her clients and Heleen has made her distrustful of kindness and eventually she gave up on the concept altogether (as the books say, better terrible truths than kind lies).
Even when she manages to sneak up on Kaz when they first meet, her request for help is presented as an offer. One that Kaz answers to, as we know- and to his credit, he's very upfront with her with what kind of job will be asked of her and the risks that come with it. The only thing he can assure her of is that whatever ugly and unhappy life she'll have with the Dregs it'll still be a better deal than living at the mercy of Tante Heleen for the rest of her life.
The thing is... Inej's offer was still, deep down, a request for help.
It's worth noting that within her first year in Ketterdam life inside The Menagerie was all she knew because the girls working there are forbid to leave; we know Inej herself tried, got caught pretty quickly and got beaten so bad she couldn't work for 3 or so weeks.
I think Inej unintentionally projects an ideal onto Kaz as a way of coping with the fact that Ketterdam itself is rotten too, that she isn't happy even outside of The Menagerie so long as she is stuck in this corrupted city and that the person who saved her, who she spends the most time with, who she works with and fights beside to, is closer to being the very embodiment of the Barrel than he is at being a better man.
Which I think is exemplified by both Kaz's ""My mother is Ketterdam" speeches and the final interactions between him and Inej after she has gone through the final stage of her arc which, non coincidentally, sees her recognizing her own gray morality by coming to terms with the fact that (although she doesn't plan on staying so she can go hunt down slavers at sea) she has learned to love the city despite all of its problems.
"I'm not ready to give up on this city, Kaz. I think it's worth saving"
I think you are worth saving.
...and if that wasn't what you were talking about then my bad😅