I do not know all the actual names of parts, but I will try to describe them so they can be identified.
I had two different tension related problems, one on a 66 and one on a 99.
1. The first picture is of a newer 66; There are two assemblies which sit in the hook ring (which is the part that rotates during the stitch cycle), in the picture the part on the left is the bobbin case, and the part on the right (with the bobbin lift button and red felt oil wick) I am calling the bobbin case spacer; I had loosened the screw marked with the arrow on the bobbin case spacer (which the manual says you should never do but which I didn’t know at the time because I hadn’t read the manual) which allows the spacer to move under that tongue and change the gap between the case and the spacer through which the thread passes; when I had it assembled incorrectly the thread would snap through this gap after wrapping around the bobbin case, because it was too tight, during the stitch cycle, but I was able to get it adjusted correctly so the spacer floated a little more; older 66s that I have owned do not have the same spacer assembly and were not adjustable;
2. The 2nd picture is of the upper tension assembly; the problem I had on a 99 was that the spring, marked with the arrow, was too tight in the assembly in terms of tension such that it would not move freely through the arc and would not move below the thread guide on the left marked with the arrow; this caused the thread, when wrapping around the bobbin case during the stitch cycle, to be too tight and then snap through gap once it made it that far; I had to adjust the spring and the assembly two ways:
a. by changing the spring location relative to the splines on the inside of the center shaft of the assembly (there is a small prong bent off the inside end of the spring which catches in the splines), and
b. by rotating the entire assembly in the housing of the machine via loosening of the set screw marked with the arrow on the right, but not really visible; it took some adjusting but I was able to get the tension spring to move freely and completely through the arc which eliminated the snapping of the thread down around the bobbin case.
This was all trial and error, mostly error as evidenced by having to fix it all upon reassembly!