IMagick is a native PHP extension to create and modify images using the ImageMagick API. ImageMagick Studio LLC did not write nor does it maintain the IMagick extension, however, IMagick users are welcome to discuss the extension here.
I do not know what wally decided to call it Imagick but it causes confusion. Imagick should show up in your php.info file and ImageMagick might show up under PHP Variables.
It looks like your hosts are talking about ImageMagick and not Imagick as you asked for.
I would contact them again and explain that they are two different things.