How do you score your GIR, Scrambling, and Putts near the green?

If I am on the fringe then I am chipping, not putting, just using the putter as my choice of club to chip with. It has also always been my understanding that once you start putting from on the green every stroke thereafter is considered a putt. If you use a different club on the green, say your wedge to tap the ball in, that is still a "putt". If you putt the ball off the green, even if you use a club other than your putter on the next stroke, that stroke and all others regardless of the club used count as putts until the ball is holed. Is that everyone else's understanding or am I wrong?
This ^^^^
 
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