@Brento,
For me the thing I see as the most likely problem is tool flex.
Both of the below contribute to tool flex.
1) tool profile
My regular-use lathe bits are 3/8", but I believe you stated earlier (or in another thread) that due to your small lathe and tool-post it was easier to use smaller tool bits. That means that by the time you grind that thin cut-off tool profile, there's really not much "beam strength" in that tool. The cross-section of the tool will only be 3/16" tall by the width of the tool you ground 0.040".
2) tool length
If you are trying to part off a work-piece that is 1" diameter then obviously you need a tool with a minimum cutting length of 0.5". From here it looks like your parting tool is probably over an inch long, but you don't really need all that length for the current operation. I understand that you'd like to have one parting tool that you can use for most diameters of work-piece that you need to cut, however it may be worth having multiple tools with different parting lengths. Start with the shortest, and then move up as the cut progresses. Of course this means a bunch of hassle changing bits mid-operation and having to very carefully align them to the existing cut.
Both of the above would be helped immediately by using an adjustable parting tool and holder.
1) Compare the cross-section of what I believe your tool looks like compared to an "off the shelf" parting tool.
The blade on the left is based on the drawing from the first amazon hit on "P1-N or P1 blade" that
@mikey recommended earlier:
https://www.amazon.com/HHIP-Parallel-Cut-Off-Blades-Various/dp/B01IAT7VXG
Neither drawing shows the taper or side relief, but they give the point.....
2) With an adjustable parting tool you can have only the small amount of stick-out you need for the current operation....in fact you can start the cut-off operation with minimal stick-out and then adjust the tool out further as you cut deeper.
Personally I have found parting tool strength to be a major factor in chatter. I originally thought that having a thinner parting blade would reduce chatter by reducing cutting forces, but what I found is that a wider blade tends to chatter less. To me it implies that the tool flex is a bigger factor than reduced cutting forces of the thinner blade.
-brino