Okay here are my beliefs and knowledge concerning cell phone records, restricted phones, and internet usage.
You can request in writing a copy of your phone records. However, that data does not usually show ping data, nor does it always show inernet usage, text messages, media messages, or messenger usage. My oldest daughter is a tech specialist for a major cell phone carrier (not Verizon). And in addition, I just had to request my own records in order to get the actual number of a phone stalker who was calling me from a private number. The records showed me all incoming and outgoing calls and phone numbers, the time and duration of the calls and that was it. No other data, no internet, no text, no pings, no messenger usage whatsoever.
For most cell phone carriers if your phone is off you also cannot use the internet unless you are using WIFI. That is you are using a wireless open/public connection to the internet. You would have to turn on your WIFI in order to make this work. As it is not set to "on" in most cases. I always leave mine set to "off" otherwise my battery goes down quickly. I can however access the internet through the phones data card when WIFI is off, as long as I have active cell phone service.
Finally- if the phone did access the internet, whether it be through WIFI or through a data card with active cell phone service then that internet usage would still be on the phone. When on the internet, the phone operates as a mini computer, and all internet data is stored on it just as if it were a computer. Deleting it does not make it go away completely, just like on a computer.
The above info is my opinion based on the fact that I was a techie in a former profession, before real estate entranced and bewitched me into its snare
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Hope this helps someone and makes sense.... lol