DCF Terminology re: "Lever"

Are these short definitions correct?

Unlevered FCF v. Levered FCF => Unlevered FCF includes interest/debt payment.
Unlevered Beta v. Levered Beta => levered beta includes company interest/debt payment.

I'm confused, because definition of levered seems "flipped" depending on whether you're talking about FCF or Beta. What's the explanation?

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