I'm surprised OpenAPI/Swagger hasn't been suggested. Define either as yaml or inline JsDoc comments if you're working in js/ts. I thought it was more popular than postman but I must be living in a bubble.
Local, open source, versioned and can serve as interactive documentation for users. Never tried it personally but it can even generate code from the specs.
My main use case for postman like tools is to get me into the middle of a workflow to reproduce some sort of a scenario. Stateless tools just don't do it, as I need to spend a long time copying things around.
Allows arbitrary scripts to save the context of requests. Hardcoded examples you can't change without committing in contract are not even close to that functionality.
Local, open source, versioned and can serve as interactive documentation for users. Never tried it personally but it can even generate code from the specs.