I did say "usually" :)
> A better distinction is that jails are run by a law enforcement agency, while prisons are not.
Depends on whether you count corrections agencies as law enforcement agencies. On the one hand, I'm pretty sure all states have distinct agencies for police v. prisons, but on the other hand, corrections officers are usually sworn peace officers and therefore would count as "law enforcement".
Another distinction, come to think of it, is that prisons are usually under federal or state agencies¹, while jails are usually under county or municipal agencies.
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¹ The only exception I've found is Chester County Prison in Pennsylvania, which is under the Chester County Dept. of Corrections. Wikipedia also has an article on a "Northampton County Prison" (also in PA), but that county's website calls it the "Northampton County Jail"; on a tangential note, if corrections agencies don't count as law enforcement agencies, then the Northampton County Jail would be an exception to the "jails are run by a law enforcement agency" rule.