Because of the recent controversy regarding pronouns on Meta, I started thinking about this question. In general, is freedom from discrimination as protected in most modern liberal states a positive or a negative right? Are there conflicting justifications for whether rights against discrimination should be honored depending on whether they are positive or negative?
The distinction between positive and negative rights is a bit misleading. A 'negative' right is merely a 'positive' right to forestall others from using their own 'positive' rights against you. Freedom from discrimination, thus, is merely the right to say: "I don't care what you think about my race, gender, sexual preference, etc; you have to treat me — for specific, delimited purposes — as equivalent to anyone else."
Going a bit deeper, this does highlight that a 'negative' right is normative (based in broad-scale social patterns) while a 'positive' right is idiosyncratic (based on personal inclinations). This is just another aspect of the intrinsic tension between the individual and the community that plays out in every social and political context. It always has to be negotiated between individuals and their surrounding community, and that is usually a contentious process. There is no one-size-fits-all answer to this question.
It's negative in the sense that Isaiah Berlin makes the distinction between removing hindrances such as chains, and, on the other hand, providing the means to achieve something, such as Basic Income proposes to do, as a positive freedom from wants (such as simple hunger or housing requirements under the notion that the Pursuit of Happiness is worthless, and therefore denied, without the necessities of life).
The larger issues you raise fall under the harm Principle of JS Mill. A claim to being harmed, e.g., by second-hand smoke, can limit the rights of others (to smoke). Being free from smoke is a negative freedom. Something is removed. In the culture wars, when someone is called a "snowflake," it means they are put down as being oversensitive. Thus, that real harm is being done to them is denied. On the other hand, most anyone who claims to be the victim of a wrong will deny that they are oversensitive, and insist they have a valid claim to the negative freedom. From so-called "hate speech" for example.