How is it different?
that's obliging inaction - it obliges someone to not say or do something which offends you - so it is a negative right
However, you are correct that the line can be blurry and right in general should be granted with great caution.
and is exactly the definition of negative rights. right there. that's the definition. that's it.
Now, you're saying that because "right to not be offended" is a bad idea, and you don't agree with it, that makes it a positive right. But your opinion of it does not matter to the definition.
The definition of "positive right" would be requiring someone else to, say, stand when you enter the room, or always address you as "sir" because you're knighted.
Let me also clarify that I very much dislike any "right to not be offended" and also that these definitions of "positive rights" and "negative rights" is clearly not very useful.
A big part of the conceptual problem is that the distinction between positive and negative rights is much less clear than people often assume. GP highlighted a pretty good example: "you have no right to offend my beliefs." It's a positive right if you call it a "right to go through life without being offended" but negative if phrased as "you have a right not to be the object of offensive conduct." And often, preventing something from happening (i.e., enforcing a negative right) may impose some sort of affirmative obligation, or impose seemingly objectionable restrictions, on others.
One somewhat fruitful way to untangle this can be to add another distinction between acts of the state and acts of private citizens. Thus, under the U.S. constitution, for example, one typically has negative rights against some state action or other (e.g., :"Congress shall make no law...abridging the freedom of speech.")
However countries like the U.S. have restrictions on those rights -- e.g. shouting fire in a crowded theatre, because society accepts that actions have consequences, and there's always a balance. You have the right to swing your fist, but not to hit me. You have the right to eat food, but not if it's in my cupboard. You have freedom to move, but not across private land (or borders).
The UN has expanded this greatly to positive rights where someone can obligate someone to do something as a right, e.g. you have no right to offend my beliefs.
In general this development is a step towards more centralization, and it is a sign of the central weakness of any centralized system which is that its limited in what it can express by the biological limitations of people that run it.