Can the terms "transcription factor binding site" and "operator" be used interchangeably in all contexts when referring to a DNA sequence, e.g. regardless of a cell type, whether or not the binding sequence is synthetic or natural etc.? If not, what are the differences?
As you no doubt know, the term operator was coined by Jacob and Monod as part of the formalism they developed to explain the properties of certain mutants in the lac operon in E. coli. In physical terms it is indeed the site of binding of a transcription factor, the lac repressor.
My understanding is that technically it is best to restrict the use of this terminology to bacterial systems in which the idea of an operon is useful. I don't think it would be wise to extend it to cover any site in eukaryotic DNA where a transcription factor is shown to bind.