According to my understanding, as soon as a 0 is in the number, the result itself should be 0 and therefore persistence should also be zero. Or am I wrong?

You're wrong, you have to multiply the digits at least once, so persistence should be 1 in that case. 0 is to be returned only when the number has only 1 digit.

You can post your code here, but using proper markdown and marking your post as having spoiler content.