My 2013 Prius C2 with a total of 4512 miles made an engine noise on start-up yesterday. The noise lasted for about 3 seconds after a cold start at 59F ambient temperature. It was a soft clanging sound and there was no engine vibration - the start was smooth. I had moved the car from the driveway to the garage the day before and shut down the engine after it had been running for about 15 or 20 seconds following a cold start. The noise occurred on the first start the next day. I also own a 2011 G3 and I am well aware that the liftback should not be shut down until the warmup cycle is completed. Since I had heard of no similar problems with the C, I intentionally shut the engine down without allowing it to complete the warmup cycle the day before. From now on, I will operate my C as I do the G3 - no engine shut-down until the warmup cycle is completed. My wife was with me in the C when the noise occurred. As slight as it was, she did ask me about it immediately!
Strange. I would look for something around the fan, belts, or flywheel pulley to see if there is contact that either moves out of the way or sets apart. As for the warmup cycle required to finish I've never experienced such in hunndreds of starts, some fairly quickly.
Mine has done this several times also, it's the conditions much like you listed. Moving the car, engine runs very shortly, and then shut off. Someone guessed that its condensation forming in the intake causing the rough run on the subsequent start up.
Mine is noisy enough that I considered mentioning it to the dealership, it's noisier than my Vibe with 100k on it, and it's a little too noisy too. I think it probably has to do with the "Atkinson-cycle" engine that I really know nothing about, maybe lifter noise is part of the plan. BTW, I am old as can be and have never had to have an engine torn down, so I'm betting that the Tom and Ray of Car Talk's old advice, "turn up the radio" is the right advise here.
I had a BMW that would be very difficult to start if I'd run it for a few seconds in a cold driveway maneuver before. Ever since that car I make sure to run any car for a minute or so if I have to cold start it.