Which one of these lamb meat parts has lowest fat -because I don't like the taste of fat-, (shanks, loin or shoulder) ? 
There are lean cuts from lamb - but you need to be very selective and expect to pay a premium. Both are cut from the back of the lamb and if you look at a cross-section you can see two roundish sections, a larger and a smaller one:  (Source) The smaller is the filet - in a lamb, it’s very thin, more like a finger or two. The second is the actual loin. They are also sold whole as longish strips or if you can’t get them, you can get a piece of the lamb back / saddle and butcher it yourself.  (Source) The key is that while you often get a chop, cutlet or back roast that’s essentially the whole piece and if you roast it whole it’s kind of self-basting thanks to the outer layer fat, the loin itself is comparatively lean and can be separated cleanly. The remaining meat is somewhere between 4% and 6% fat, according to my research. That’s so lean that you really have to be careful or you end up with a dry meat. If done correctly, it’s incredibly tender and juicy and has very little to none of the “lamb fat” flavor. Shoulder and leg are fattier to begin with and the meat is more marbled. You can’t remove the fat prior to roasting and it will always “seep into” the dish. 