I see the first three concepts used interchangeably (as far as I understand)...

Either you misunderstood, or the people using them misunderstood.

Lambdas are anonymous functions. In javascript terminology:

    _.map(someNumbers, function (x) { return x +1; } /* <- lambda */)

Closures are a way of hiding state (or at least data) in anonymous functions:

    makeCounter() {
        var x = 0; 
        function numCalls() { return x++; }; <---the variable x is "closed over"
        return numCalls;
    }

Monads are a method of encapsulation, with goals similar to object orientation. OO is about hiding data inside objects, whereas monads are about hiding most of the outside world from your functions.