How to trace a private number when it leaves no trace on call-log (call register) of the mobile phone? Even no detail is found for that private number by the police when they acquired the Call Detail from the mobile company.

If the police cannot obtain the information, neither can you.

Whether an unlisted number and private number a same thing or different?

Both, unlisted is a phone that a number is actually listed some place, many phones are no longer listed. In them, the phone is set to privacy, not to give out the number

Whether an unlisted number and private number a same thing or different?

Doesn't matter, neither one shows up on call display.

And very odd if the police can not even find evidence of the call.