As far as I understand it (and I could be easily and completely wrong) "sprezzatura" is a way of dressing oneself without putting too much thought into what goes together well, or at least give the appearance of not having put a lot of effort into it. The latest version, however, seems to have lots of little things like wrist bands and accessories in general incorporated, while leaving certain details unattended. One of the latest trends, for example, is not to button the collar of a button down shirt, roll up the coat and shirt sleeves(Miami vice style!) and other variations.
The menefreghismo part is exactly what Jeffery wrote: "I don't give a shit!" Which in this case means that whoever made this coat (perhaps even along with the customer and the cutter) didn't care much about the make of that coat.
The faults Jeffery mentions and shows are indeed so obvious, that they could not have been missed by any tailor/ cutter or even customer.
Or is this just sloppiness, certain tailors' taking advantage of clients too ignorant and obsessed by status to care?
That could be part of this. For some time now it is considered "rude" (to put it mildly) to critisize certain tailoring houses, especially in Naples. Although this type of execution is very hard, if not impossible, to defend.
"Nur der ist Meister seiner Kunst, der immer sucht, das Gute zu verbessern und niemals glaubt, das Beste schon zu haben."
"Only he is a master of his art who always seeks to improve the good and never believes to have the best already"
http://www.dressedwell.net/ It's snarky, but fun.