I find it interesting in how language evolves throughout time and regions.
I've read that the original term that many of these get attributed to is "pottering" which the Oxford Dictionary indicates came from the 17th century verb "pote" which means to kick or paw at something repeatedly. From there it came to mean to occupy yourself in a pleasant but ineffective activity or to move about aimlessly. The "putter" variant shows up in literature in the beginning of the 19th century.
Putzing shows up in the mid 20th century and is believed to be a variant of the Yiddish term "futz" which means essentially the same thing.
Piddling comes from "piddle" which has changed in meaning throughout the years. In the 1600's it meant to pick at your food while in the late 1700's it took on quite a different (biological) meaning. As an adjective, "piddling" is usually in reference to something relatively small, usually monetary, like "a piddling 2-cents".
Dave
PS I am usually doing these puttering, piddling, activities in my shop on some kind of thingummy, thingamajig, or thingamabob.