Tailstock:
Looks like you have a long extensions(s?) on the tailstock. That live center looks out of proportion. Try a short dead center for more stiffness. Keep the tailstock extension at a minimum and locked.
Steady rest:
That steady looks very light, and is located fairly far from the cutting location on the workpiece. That circular base makes zero sense to me. I'd make a new steady (they are not hard to make) that is solid, and keep it close to where you're cutting.
Workpiece: I assume you weren't cutting the thin end first as shown here, but did that after turning the midsection. Part stiffness and order of operations is important. But you did note that you had just set this up with a finished part.
Insert:
I can't tell anything about the insert from the pic (all pics are quite small). But again, I'd suggest using a regular hand-ground lathe tool of the largest cross section your toolholder can take. Probably 1/4" but if it can take 3/8", all the better for stiffness.
Toolholder:
That toolholder puts the location of the point of the tool far from the center of the toolpost. In other words produces a long overhang. I don't know if that is original to the Pultra. I'm guessing not.
Very small lathes are not improved in stiffness by miniaturized typical quick change tool posts. These tend to reduce lathe stiffness and increase twisting stress on the carriage and slides compared to simpler traditional toolholders. They are really suited to larger lathes.
I think an ideal toolholder system for a small lathe is the "Rose" type, which appeared quite awhile back in Model Engineer. Invented by a Dr. Rose who actually lived in the New Orleans in the US. These are very simple to make, and have the greatest stiffness possible on a small lathe.
Many of the modern "must haves" for lathes: insert tooling, quick change tool posts, live centers, and three jaw chucks are counterproductive for small lathes where stiffness is the greatest concern. Faceplates and collets, HSS tools, dead centers, and a simple short traditional tool holding system work better
The name of the game here is solidity, stiffness, short overhangs everywhere, sharp tools, and proper feeds and speeds.