I still think the Lady Day Coat from j crew is the best for petite, thin women. It's classic, subtle, and not overwhelming...and it also comes in petite sizing. Mine is ~14 y.o. Quality has gone down a bit, but still good I believe.

https://www.jcrew.com/p/womens/categories/clothing/coats-and-jackets/topcoat/new-lady-day-topcoat-in-italian-double-cloth-wool-blend/BM964?display=all&fit=Classic&color_name=holiday-red&colorProductCode=BM964

I'm similar build to you, OP. I go for black wool trench coats with a belt and just above the knee in length, and usually have them tailored so they are well fitted. Macys has a good selection online - I usually order several to try out and once I've picked a style I test out a size up and a size down to see which works best.

Oh no, heaven forbid someone thinks you look FAT!

I'm not gonna say my age because I don't think style should revolve around age. I am petite and thin. What coat or jacket would be recommended? I have a big puffy polyester jacket with a hood but I dont like how it makes a swishy loud noise. The polyester is nice for the cold but makes me sweat indoors. Also it makes me feel and look fat.