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:
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.
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.
As someone who also hates fatty meat, with lamb honestly it doesn't matter.
It's very fatty meat, all cuts.
My local supermarket's "low fat" lamb is still 20% fat. They don't tell you what the rest is.
The trick is to cook it for a long time - 4 hours or so, or all day in a slow cooker. The fat doesn't magically disappear, but it gets rendered down into something far more tasty [& incidentally, if you're like me, a whole lot less 'gribbly' to chew;) Far more palatable. Cook it long enough & you really can't tell the fat was there.
That's how any good 'curry restaurant' manages to make lamb curries that taste great & don't feel 'fatty'.
To add to the already good answer, you are not escaping fat when you eat lamb. It is also an incredible rich fat. Especially the fat around the ribs. This is why currying your lamb is usually a good thing as it just tempers the richness so as to make it palatable. Unfortunately though if you have a constitution that cannot handle rich food then lamb is just simply not going to be good for you.