This article is totally misleading; watch the video.
She was nodding at the president of Hunter College (Jennifer Raab) when she said “Madame President...that has a nice ring to it...”
She was not referring to herself. (Although, her comment makes it clear she STILL hasn’t gotten over the thumping she received.)
>>>She was nodding at the president of Hunter College (Jennifer Raab)<<<
My hunch is that Raab jokingly, or fawningly, referred to Hillary as Madame President in jest . . . and Hillary was simply acknowledging it. That is usually the problem with small snippets of a larger event.
Yes. You explained it well.