The term used [about trump] “birther”-is very derogatory and is meant in a derogatory way.
I'm not sure anything can be derogatory to Donald Trump, although I would concede that as to him, the term "birther" is probably misapplied. I'd be inclined to reserve the term "birther" to someone who sincerely believes, in the absence of any substantial evidence, that Barack Obama was born in Kenya or on the moon. Distinguish this particular disability from the attitude of any sociopathic narcissist willing to embrace any idea, no matter how fanciful, and no matter how thoroughly he holds such idea in contempt, with the purpose of incurring favor among the most gullible of our polity (whom he also holds in contempt). Call him a "trumper."