A trench coat is exactly what you described. But there are variations on that theme that Irene pointed out.
I have a 12 year old Jones New York rain coat that's water-resistant, khaki colored, but has a shawl collar that blends right into a hood (perfect for the rain). There are no visible buttons down the front -- although it is single-breasted, it's also full-length, a-lined, and has no belt.
I consider it a trench coat because of its function (a dressy rain coat). I could be wrong. However, it has been in my closet for 12 years and counting because it has the versatility of a trench coat for me without all of the visual distraction that doesn't work for my body type. I hope that helps.
