Slightly nitpicky, but is something a PaaS if you run it yourself? Anything as a Service isn't a service if you're running it yourself. It's just ... infrastructure.
”Service” to me is just ”delivery of something with a specific scope and a defined contract”, not so much about who delivers said ”service”. Many IT depts would do themselves a massive favor to deliver actual services instead of “just infra and some stuff thrown on top” and call it service delivery. Tools like in this link can help, but a big part is simply about automation and delegation/self provisioning.
Platform as a Service or anything "as a Service" means someone else provides it as a service (ie subscription). The Platform part is all this is offering. So it is not a Platform as a Service.