I just did exactly this. There is now ten miles a week of walking in my commute. I wore out three pairs of shoes before spending a bit more for some Doc Martens; they meet exactly this requirement. Not too heavy for "city" walking, although a bit heavier than trainers. No problem in the wet. Durable. The only reason I wouldn't go hillwalking in them is wanting to keep them clean.
(This is almost literally the "Vimes boots theory of poverty" from Pratchett)
I've used the same pair of lightweight waterproof Timberland boots for the last 10 years (in the Spring, Fall, and Winter). It's not a problem at all walking around a city (Boston & NYC) in them, or hiking outdoors. In the summer, I go with my barefoot shoes, but that's a luxury and not necessary.
I've effectively used hiking boots for city-life for years. Yes, they are heavier, and the first week you get tired of walking faster than everyone else, but then it becomes natural.