intrinsic concerns the nature of the thing/person referred to

"immanent" is mainly applied in philosphy, most often in conjunction with a specific system/cognizance. However, nowadays it might concern concrete systems as well (political, technical...)

intimate = "inneres, geheimes" usually refers to some kind of emotion

organic = organisch concerns a concrete organism. The right translation might, however, be "immanent". For example, an "organic fault" is usually a "systemimmanenter Fehler".

There's really no one definite translation for these terms; they all depend on the context.