Which one is important is going to depend on the context for sure.
If it’s an open source library, they probably won’t care about 1.
If you’re working on internal software used by other developers within the company, management probably really does care about 1 because it’s going to impact their timelines.
If you’re working on a proprietary user-facing API, then even if it doesn’t cost your company anything management might still care because it could piss off valuable customers.
I think that, for what ever decision OP is trying to make, looking at that context is more important than quibbling over what exactly constitutes a “breaking change.”
Which one is important is going to depend on the context for sure.
If it’s an open source library, they probably won’t care about 1.
If you’re working on internal software used by other developers within the company, management probably really does care about 1 because it’s going to impact their timelines.
If you’re working on a proprietary user-facing API, then even if it doesn’t cost your company anything management might still care because it could piss off valuable customers.
I think that, for what ever decision OP is trying to make, looking at that context is more important than quibbling over what exactly constitutes a “breaking change.”