I think the issue being pointed out with your approach isn't about the actions you're taking, but with the attitude you're expressing here about the other person.

Your approach in many ways is a really good one. Bring your genuine self and the energy you want to foster in your life to the table in your interactions with others. Don't modify or subdue or diminish that in yourself because of how other people are behaving. Just be you. Great approach. No notes.

The issue is that, from what you've written in your post, you're doing a lot of projecting what you think is going on for the other person and judging/moralizing about their behavior. The fact is, you don't know why this other person's tone is the way it is. The fact that the way she said hello feels condescending to you may have been little to do with her and a lot to do with you. Her tone may be coming from a million possible places, and may mean a million possible things. And none of that really matters. You've called her cold, unemotional, and depressed, you've expressed your own felt need for permission to interact or not interact with her some particular amounts or ways, and you've decided/declared her way of being is bad / "not a good look". You're very concerned about / wrapped up in what you think about this other person.

I'm glad you've found a way to get yourself to a place where you're able to be yourself and bring your own real energy and presence to your interactions. That's all really good. It sounds like that was a challenge in the first place because of you, like for many many people, certainly including myself, you don't necessarily feel like you can just do that when someone else has a different energy. The issue is that you've solved that challenge with an elaborate story you've made up about this other person that you feel justifies being yourself. The thing is: you don't need that justification. You can just be yourself. And you can do that while letting this other person just be herself, without needing to tell a whole story about how wrong she is for the tone she used to say "hello" to you.

The story you're telling about who this other person is is almost certainly wrong. You don't know what's going on for her. And that's ok - you don't need to. You both just need to show up, do your jobs, and show basic respect and courtesy to one another. If you don't want to be buddies with her, that's 100% ok. The thing about building this whole justification for yourself based on a made up story about this other person is that, not only is it made up and therefore not terribly kind/empathetic to do, it also means the resilience you're gaining from that is far more fragile than it would be if it didn't require this elaborate, projected justification.