The first issue is whether there is something about the surface between the ball and the green (or even the hole) that has a substantial risk of knocking the ball off line. If yes, chip, don't putt.
The second issue is whether there is much grass directly behind the ball so that the flat putter face will meet resistance trying to contact the ball cleanly. If yes, chip, don't putt.
The third issue is whether the grass between ball and green is too thick such that you would really have to KILL the putt to get the ball all the way to the green, in which case you're not likely to have good line or distance control even if the first issue above is okay. If yes, chip, don't putt.
The fourth issue is whether given the substantial distance from ball to green and to hole, you are better on distance chipping to a spot and controlling the roll-out versus putting the total distance. If yes, chip, don't putt.
The fifth issue is whether there is something about the surface at the ball or the lie conditions such that chipping is "counter-indicated" as the doctors say (e.g., very packed, dry hardpan, or shallow wet bunker sand, etc.) but that does not also make putting more risky (and all the above 4 issues do not "counter-indicate" putting). If yes, then putt, don't chip.
The upshot of this is that chipping is typically the default decision, and putting from off the green gets a green light in special situations: short, tight grass with uniform true surface between ball and green or hole and the distance is not too great. Even then, the choice depends upon whether you can do better in the specific situation with the putter or with a chipping club. This in turn is somewhat dependent upon your practice and your preferences.
Additionally, the "Texas Wedge" is putting from off the fringe with a bladed iron (usually a wedge, as this has a nice thick bottom edge and pretty good loft so the shaft does not have to be angled too much forward if any), the idea being that the wedge's edge (lovely rhyme, that) will cut thru grass behind the back of the ball better than the planar surface of the putter face (another rhyming gem). The "Three-wood putt" is sometimes used when a little loft across the fairway is not all that harmful, when a little greater distance is involved, and sometimes to get the ball up out of deepish grass and rolling along when a Texas Wedge's leading edge would not suffice to get the ball up and out.
Finally, there are occasions when the putter is the right club for coming out of a greenside bunker. If the sand is not too puffy and there is no big bunker lip to traverse, but there is a reasonably smooth transition from bunker to green, then the available choice is whether to play the full bunker shot or the putt. And sometimes even when there is a lip to traverse, the putter may still be the play if the bunker is on the short side so there is not much green to use with the normal bunker shot and/or the green slopes severely away from the bunker. These weird (i.e., "creative") uses of the putter send the ball onto the green with more distance control than the land-bounce-release action of the normal bunker shot. For example, if the green slopes away and the bunker is short side with little surface between bunker and hole, putting the ball up and out of the "ramp" of the bowl-like bunker's forward wall may appropriately launch the ball gently into the air to land softly on the surface just past the fringe and then amble on down to the hole!
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