Based on your definition, then any static analyzer is a type system, because type information and the usage of those types is basically all that’s available for static analysis.
Types define what operations can be performed. The borrow checker looks at allocations and determines when they can be freed. It really has nothing to do with types. An ideal implementation of the borrow checker could be totally type unaware and work with dynamically typed languages.
Probably you could build such a thing without thinking about types per-se, but I don't think the designers of Rust did that, and I am not sure that makes it "not a type system" anyway.
> any static analyzer is a type system
"Any static analysis checking correctness is either a linter or a type checker" isn't a claim I would make, but I am not sure offhand how I would argue against it.
Literally all type systems could be described as “a static analyzer” that tries to assign and validate properties over the code it’s analyzing.
All compilers also rely on the results of that static analysis to direct codegen.
Rust’s type system implements substructural typing, and the borrow checker is an integral element of that type system.