I'm not aware of any secondary pumps for the turbo. There are intercooler pumps, but that's not going to have any effect on turbo temp. Usually, flow is maintained after shutdown via thermosiphon.
Yes they are temp controlled, but with a max time. It's the radiator fans. People are taking it literally. Turbo timer is a generic term because the function is the same. You'd notice all the ecoboosts out there idling after the driver gets out if it was literally a timer before allowing the engine shutdown. There are all sorts of implementations of protecting the integrated turbo.
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