[Task]: Refactor Error handling in Rust/TS

### Feature description

Each fallible operation should have its own error type tailored to model it.
Deprecate (or remove) the old `Error` type and refactor error-handling for each fallible operation our library provides.

Write errors that leverage `std::error::Error` functionalities and ensure they are as modular, correct, and informative as possible without exposing any unnecessary foreign type. See [Sabrina's blog post](https://sabrinajewson.org/blog/errors) for directions.

### Motivation

A single huge `Error` enum for the whole crate doesn't cut it and it hinders the composability of our library's code, besides making it almost impossible for our users to recover from those errors.

### Requirements

Refactor the `Error` type into more modular errors

### Open questions

_No response_

### Are you planning to do it yourself in a pull request?

Yes