Forgive my ignorance, but are Woolrich Woolen Mills and Woolrich John Rich & Bros related companies? As in is it an Arc'teryx/Veilance situation?
Yes, however the first link isn't Woolrich Woolen Mills, it's just plain Woolrich. I mean you could argue it is since Woolrich is a "woolen mill," but they are different brands of Woolrich.
Woolrich, which your first link is, is just plain old Woolrich from what I remember as a kid. It's the cheapest level of Woolrich.
Woolrich Woolen Mills is the mid-range brand of Woolrich. Pricing is higher and more than "mid-range" depending on your take. Mark McNairy currently designs Woolrich Woolen Mills and prior to him it was Daiki Suzuki of Engineered Garments. Designs tend to be the most adventurous of the Woolrich brands.
Woolrich John Rich & Bros, otherwise also known as just plain Woolrich, is the highest end Woolrich. I've also heard it referred to as the "Italian Woolrich". Their site only sold to Europe until about 12 months ago. The parka in your second link is Woolrich John Rich & Bros. I have a few pieces of outerwear from them and I generally like them. Designs aren't too adventurous, more "heritage" inspired I'd say. Tokihito Yoshida of Barbour Beacon/Barbour Tokito fame has done some of the design. I'm pretty sure he did after he left Barbour too, but to what extent I don't know. (In fact I'd be curious to know what he's doing if anyone knows). I absolutely love their physical label with the sheep.