I'll confirm what Scott said. Putts from the fringe or fairway do not count as hits.
I had a thin red line on a Nike L71 this morning so I ran a test on the "Play one hole" option choosing Chambers Bay 15, a par 3. I zoomed in on the Hits indicator and took a screen shot before and after each shot. I then compared each hit indicator by zooming in again in a photo shop program. I could clearly see when the hit indicator changed.
Here's the results:
My 4I tee shot - hit used (no surprise there)
Putt on green - no hit used (no surprise)
Putt from fringe - no hit used
Putt from fairway - no hit used
Chip on green - no hit used (I was surprised)
Pitch on green - no hit used (I was surprised)
Flop on green - no hit used (not surprised by now)
Punch on green - no hit used (no surprise)
Full shot on green - no hit used (no surprise)
Chip from fringe - 1 Hit used (no surprise)
Mulligan on green - 2 hits used (no surprise)
Pitch from fairway - 1 Hit used (no surprise)
The pitch from fairway was the last hit on that ball, I had 5 hits left then when I started. The ball was replaced after that pitch, and I was on the green with putter.
As Scott indicated, safe to imagine any shot with a putter does not count as a hit.
Also it seems safe to imagine any shot with any club on the green does not count as a hit.