A.C.
Polyurethane comes in both water-based and oil-based formulations...so they may be talking about the same things just using different language.
An oil-based finish is generally going to be more durable and higher gloss finish. The odor is usually stronger and requires heavier chemicals for clean-up.
Water-based dry quickly, have lower odor, have more finish (satin, semi-, gloss) options. They are not generally considered as durable, but professional grade products do a great job.
I wouldn't mess with oil-based DIY...but would go for it with a professional if you want a higher shine finish.
P.S. I am jealous...I SO want to get my floors redone in my 1941 home.