managed generally means that the datacenter/provider also takes care of OS upgrades, installing patches, installing software as required. unmanaged generally means hands off except for hardware related tasks, and generally if it is the fault of the sys-admin then they might charge and hourly fee for it.
So can it be possible that some providers use these terms interchangeably? One says "it's managed because we take care of the hardware", the other says "it's unmanaged because we don't take care of the software." Seems confusing to me.
I definitely understand why you would want to take responsibility for OS upgrades and the like.