Does verison charge for personal Wi-Fi use?

I'm on a plan that has limited data usage. I thought that if I turned off the cellular data on my iPhone and use my Wi-Fi for Internet then they won't count it but my mom called them and they said that personal Wi-Fi also counts as usage. I'm finding it really hard to believe this.

Also, my mom got me the plan and she told me I have half an hour of Internet a month which also dosn't make much sense. She got the plan as a group thing in her business, if that's somehow relevant.

I find that hard to believe. Although I'm on a family plan, not a business one, I could never see Verizon charging for Wi-fi usage. Mostly because they have no way of being able to tell that you're using Wi-fi. If you're not using their 3G network, it should not be a big deal. Even if you are using 3G anyway, the lowest plan is 2 GB of data (what I have) and I never even use half of that in a month without ever using Wi-fi. You should be fine, especially if you use wi-fi.