Since there is some ambiguity, I'll describe the circumstances that would make me prefer one option over the other 2:
Some obstacle is present along the path of a putt (sprinkler head, tree root, another ball, or something like a kidney bean shaped green where the putt couldn't make it to the hole without leaving the green again)
Some kind of maintenance problem is present along the path of a putt (turf damage, heavy ball marks, poorly cut edge of the green that could deflect the ball unpredictably)
Anything over 3 feet off the green or so
Basically just any unusual conditions that would increase the chances of a putt bouncing off line by a large amount. If it's just flat/smooth fairway for less than 3 feet and then a well-maintained putting surface beyond that I won't chip because it has a worse result, on average, than putting. I should probably putt from longer distances off the green, but I haven't practiced it enough to feel comfortable with my distance control unless it's a very brief stint rolling on the fairway.
If I'm up against the collar of the rough I'll use a bellied wedge to "putt" the ball without having interference from the rough. Sometimes I will experiment and use a bladed 3-iron or a 3-wood chip instead, but I'm mostly comfortable with a bellied wedge here since it pretty much perfectly skims over the rough.
Anything not listed above, which includes almost every shot where the ball ends up on the fairway within 3 feet of the green.
That certainly makes sense, but as I understand the Rule 1.2, the DQ penalty would to be assessed by the Committee for the serious misconduct. The Committee also has the authority to choose to warn the player, rather than apply the DQ penalty. In real time, it seems unlikely that a decision to DQ the player would be made while the hole is still being played. In real time, and in match play, the smart thing to do is to concede the putt before it can be played.
I have used a putter for a ball mark. I have also used it as a measuring guide to move my marker, off to the side, out of another golfer's putting line.
I have also used little stones, broken tees, bottle cap, and even my divot tool.
The only "iffy" thing I ever saw was on a dew covered green. The guy used the left over ball trail as his mark. He replaced ball where the trail ended
My marker of choice is high dollar poker chip from a casino that closed down several decades ago.
My two PWs have a "P" stamp in them. Both are 48* degrees, purchased 10 years apart from each other. Same brand/company, but different styles. They both play the same for me
That said I have seen some older clubs in the 46*-49* degree range, with the number "10" stamped in them.
Wedges, or irons, it doesn't really matter. They are all just tools, with a specific use, based on the golfer's talent level.