For the former, you're absolutely right. For the latter, multiple-pass overwrite provides greater assurance than single pass. Following that up with physical destruction of the platters provides further assurance.
Again, thread modelling is crucial. There may be no point wiping!
The required equipment makes magnetic data unlikely to be recoverable except in exclusive cases where certain government agencies with sophisticated labs get involved, but in short, storage leaves lasting (trace) physical changes behind.
SSDs and operating systems without proper Secure Erase support can very easily leave behind complete data.