The defocused effect on some of the rain drops is because they were closer to the camera and out of the plane of focus. I was using my lens hood, so the front glass stayed dry. These shots were with my long lens (the 70-200), shot from about ten yards away from the subject, so there were plenty of rain drops between me and the subject. But only the ones in the "beam" made by the flashes were lit up.
If the flash is your only light source, it doesn't really matter how big the light source is; bare flash or umbrella, it's going to freeze time. The flash duration is extremely fast, it will freeze motion similar to a 1/10000 shutter speed (more or less - some flashes are faster or slower depending on how long they take to "cool down" after the fact).
You could probably get a very interesting streaking rain effect with a combination of ambient and flash. You'll start to see raindrops as streaks instead of dots around (approx) 1/400 or slower... so if you have a continuous ambient light source, a relatively long shutter speed (like 1/10), then pop the flash at the end of the exposure (second-curtain sync), you'd probably see streaks of rain with glowing globes at the end when the flash fires. I haven't done this but it sounds like a fun project.
Or, just put bags over your gear and go out and experiment. Get a couple of Optech Rainsleeves for your body and lens; get some Ziploc baggies for your flashes, and go out and see what happens.