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We have had several plays where a ball is hit to shallow center field and the batter advances to first without a throw out attempt. (IE: the CF fields the ball and returns it to the cutoff who returns it to the pitcher.) Can anyone direct me on the proper way to score this? (Not sure if I use the term 'advanced by batter" when it was the batter himself who hit the ball or should I use "defensive indifference" or some other way. Please advise.

I just use "held up" for the batter.

Yes, "Held Up" is the way to go. "Advanced by Batter" will be treated the same way statistically as "Held Up", but you would usually use that for the runners, not the batter.