No, they are not interchangeable: if you read the definitions that you have provided, they are clearly different. That said, I disagree with the definitions that you have quoted.
Talk down to somebody means speaking to somebody in very simple terms, as if they are a lot less intelligent than you.
As far as I'm concerned, it's not at all about acting as if somebody is less important than you. As anotherdave commented, a good expression for that is looking down your nose at someone.
putting somebody down means saying or implying bad things about somebody, usually in front of other people. It often involves criticizing something that they have just said, but can also involve saying or implying that they are lazy, stupid, unfit, unattractive, boring...
As the definition you quoted says, put somebody down is informal, and definitions of informal expressions are sometimes less precise in their usage and interpretation than more formal expressions.