I’m not a shopper. I decide what I want, and then go to get it. So I don’t frequent places that might have a nylon windbreaker. Where can I get an ‘old school’ one? Specifically: Unlined, non-rip-stop, zipper front, slash pockets. I’ve seen a couple of snap-front jackets online, but I’d like to find a zip-front.
Dude, the Eighties are calling you. They want their fashion sense back. 
Sorry, no advice. I hate to clothes shop almost as much as I hate drinking lite beer. Maybe REI has something you could use at only four times the cost that you used to pay for clothing, back before it became “outdoor gear”. 
Stranger
'60s, actually. You may be thinking of Members Olny jackets (which do, unfortunately, seem to be making a bit of a comeback).
Here. Can’t tell if it has pockets, though. Oddly enough, it’s “made in the USA” AND imported. To a US site. I have no idea how that works.
Or, if you could get over the anti-ripstop bias (what’s that about?) this one is perfect.
I recently needed the same thing. Unlined with no sports team’s or manufacturer’s logos. Damn hard to find.
I found a windbreaker with the silly Member’s Only style straps for $30 at JCPenney & settled for that. Zip front, unlined, internal breast pockets & external slash pockets. perfect except for the decorations.
Looking for it now I see their website has stopped showing a lightweight jackets category but has 3 new categories of winter weight stuff. Oh well.
The first one looks pretty close, only I was looking for white or dark green. The second one is available in green (but not white). There are a couple of details I don’t care for: the diagonal seams on the shoulders, and the upper pocket. But it’s not bad.
Try http://www.campmor.com ?
I have a navy blue one - full zip, pockets and hood, I’m pretty sure they’re still available in shops here (Ireland)
Windbreaker jackets are made only by Windbreaker, Inc. — it’s a trademark.