To answer your question: No it does not start counting when the Master switch only is turned on. As you said, the oil pressure switch has to be sensing (I don't know how much) oil pressure to activate the meter. In the R44 the meter is activated by a micro switch mounted on the collective and closes as soon as some collective is pulled. This is called a collective hobbs.
However, if in your aircarft the meter is starting with only the master switch on, then you are beeing ripped off. If so, I would confront the owner and ask for a refund of over charged flight times.