The simple rule you provide doesn't describe any real system; it seems to be an approximation drawn from how felony charges under state law in the US usually work, but more generally, in the US, Jails are local (usually county) facilities that both house suspects awaiting trial on state charges and convicts serving state misdemeanor sentences, while prisons are either state facilities that hold felony convicts or federal facilities that can house anyone from pre-trial detainees on federal charges to convicts on any kind of federal charges.

There are some variations among states, too; e.g., in California, some state felony convicts serve their sentence in county jails rather than state prisons.