Safety is the whole idea behind Rust, and if you draw the line here, that's neither in line with the Rust ethos, nor particularly valuable. After all, code that "expects to handle" null was pretty rare too ;)
Safety in Rust's context is UB-free, memory errors-free, data race-free.
Safety in software engineering is more about designing systems with some degree of assurance against certain failures, but not about surprises or expectations of a programmer.
The usage you call traditional is perhaps common, but not really rooted in anything in software engineering. I'd call it an informal meaning, maybe.
Safety as in unsafe blocks means what you say. But safety as in why so many people use rust is actually what arcticbull says. I think this is something that gets lost on a lot of people who don't use rust. The first form of safety is properly called memory safety.
Rust as a language isn't just designed to avoid undefined behavior, it's designed to make you write correct code, where correct means it does what you want it to. Obviously rust doesn't always succeed at that broader goal, but it actually does a pretty good job all things considered.
Arcticbull's description of rust's ethos is spot on.
It is a common misconception to conflate safety with functional expectations. A program that only calls panic() is perfectly safe.