The L19 spider is not a great putter and the L71 is. The advice on balance / precision I concur with fwiw.
EDIT: you def. do not want the L92 ding or bust using the starter rock though IMO. Great putter, but especially not suited to the starter rock that's all.
Personally though I would look at why I am not close with my approaches often enough. I would consider a better ball (still cheap) than the starter rock if possible (assuming the rock ball is what you often use - never checked much). Then I would roughly map those irons / new ball combo (low cross wind Beth 10, 11, 12). Once that is under control and my short wedge shots are bango (see forums) I would think again then.