iPhone 5 confusing Wi-Fi data with Cellular?

I bought my iPhone 5 just a few hours ago. When it was set-up, naturally it used 4G to access the Internet. Once I got home, I connected to my home Wi-Fi.

It is my understanding that the iPhone will always prefer Wi-Fi to 3G or 4G. Plus, the entire night while I used it, I never saw anything other than the WiFi symbol in the corner.

I got a text message from Verizon saying that I had used 75% of my allowed data. (2GB on the plan, but it is prorated) I checked MyVerizon, and it said I had used 427 MB of Cellular Data!! I clicked a link to open a detailed chart, which informed me of only four instances of Cellular data, which totaled up at 1.8 MB. This is a number that makes sense.

So I turned off Cell Data entirely, but left the Wi-Fi on, and continued to download videos from YouTube. Moments later, I got another text from Verizon saying I had used all my prorated data! I checked and Cellular Data was still off! The note even said "With this turned off, all data will be restricted to Wi-Fi"! So I clearly didn't use Cell data, but Verizon thought that I did! Now I'm over my limit and possibly in huge trouble.

The figures given seem consistent with my overall usage, though 99% of this usage was on Wi-Fi. I would stake my life on that fact. No joke. Is it possible that my Wi-Fi data is somehow being confused with Cell Data, even though it's turned off?

In the morning, I'm going to visit the store that I bought it from. However, any info that you guys can give me would really help.

Thanks.