Some coaster brakes use oil they are not all built the same New Departure have a row of discs inside
and should have some lube the Sachs and Perry's have brass brake drums .
That Schwinn guide probably shows how to service all of them .
I've done CCM Model 37, Perry , New Departure , Shimano, Sturmey Archer, and a bunch of Generic and European hubs and there's all kinds
and they all work a little differently .
The hub in the picture has no markings on the brake arm or shell so It could be a Sachs or a Perry
but the axle actuator and brake drum out of it are now in my 80 year old Perry hub shell on my bike .
And it left 2 skid marks on my neighbours driveway so it works .
I'm just telling you what worked for me .
Also told you my version as not everybody has a vice on there workbench or even a workbench .
Let me know how you make out putting it back together and I'll help out if I can .