[Blue37]

Let me rephrase your question to see if I understand it.

If the clock is stopped by the officials to attend to an administrative task such as administering a penalty or moving the chains on a first down, does it start when the referee gives the ready-for-play signal or does he also have to give the start-clock signal before the clock starts?

If that is your question, I would answer this way. A good crew will let the clock operator know the status of the ball and he will start the clock on the referee's first movement. (edited to add: assuming the clock operator is on the sideline. If he is in the box, there is no way to tell him other than the signal(s).)

You also asked about the necessity of giving both signals. Some referees give both and some only give the start-clock. I give both because that is what my assignors want me to do.