For a hilly, rainy winter commute you need:
Sufficient tyre clearance. If you want winter studded tyres you need more room.
Threaded eyelets for bolt-on fenders (SKS chromolplastic)
Eyelets for a rear luggage rack.
Selection of gears inc low ones.
Optional but very nice to have are disc brakes. Mechanical ones are fine, you dont need fancy hydraulic.
Bad to have: any suspension system
A standard hybrid style bike will do the job well and is cheaper and easier to find than a touring or cyclo-cross style road bike. Hybrid size tyres are about 35mm-38mm and most models have room for wider studded tyres. You can always fit narrower one. For being on time rather than being the fastest, it it hard to beat Schwalbe Marathon Plus, the toughest , most resistant model.

Budget for rack, fenders, luggage, lock, lights, helmet. You can use any waterproofs you already have.