Count me among those for whom "private cloud" effectively means a colo solution (a sibling comment[1] mentions another version of this, which I would also consider a private cloud since it's on dedicated hardware). I would call what you first referred to as a "managed cloud" service.
Where I work we refer to a product offering as "private cloud" that means we will run the software on their own AWS account. This keeps their data separate from other customers and allows extra security(restricted access to US employees, restricted access via VPN, etc).
I'm also an ex cloud engineer(as in building, operating, etc) so I understand the hosting meaning of "private cloud" to mean a cloud under control of the user(software and hardware), such as a company running an Open Stack install in own datacenter or colo.