The purple coat looks fabulous on you!
It's too bad about that closure. It's a great cut for you, but impractical, I agree, if it only has one button. (People, what are you thinking???)
I can vouch for the quality of the J. Crew coats. I have an older one -- the shorter version. It wears like iron and is warm and quite wind-proof because of the weave.
J. Crew makes a version of this coat every year and this year they are also offering the longer one, which you are trying. I am not sure if the cut is the same as on mine (the shorter version) but in the shorter version, the cocoon coat has (I think?) a version of a raglan sleeve (or some other sleeve name I don't know, since I'm not a sewist)...in any case, the sleeve is not set in at the top of the shoulder but the fabric extends all the way down the arm; however, unlike some raglan sleeves on the underarm side, it is cut quite high (i.e. more like a traditional set in sleeve on that side of the body vs. a loose fit beneath the arm.) The coat is a bit wider up top and narrows to the bottom. This style of coat can be great and easy to wear for those with an inverted triangle body type or a rectangular body type -- it might be a bit less comfortable for a pear shaped body type, depending on various factors. One of which Janet mentions -- the zipper needs to be a two way zip. I am very grateful for the two way zip on my coat -- when I am sitting, I need to unzip it, and I also feel more comfortable striding if it is a bit looser at the bottom.
So test it out in motion and do a sit test if you think you might keep. As others have said, it has a slightly more casual feel because of the cut/ shape. But it looks nice from what I can tell and it really is a quality item.
Nice of your mother to want to keep you warm! But I also agree with others, you shouldn't keep anything you're not super happy about, because you won't reach for it.