With quick change tool posts, there are 3 basic types of holders for parting tools.
The most common is the "Number 7" one that's included in most QCTP sets. It holds a HSS blade at an upward angle, and they are such a pain in the neck to keep adjusting the center height as you resharpen the tool. I swear those are among the worst holders, so fussy to get a good cutoff. When you finally get a blade sharpened in a way that it works and you don't get the crunching climb-up or chatter, you're so often reluctant to resharpen it because you're afraid it'll never work the same again. And usually, it doesn't.
Next you have the straight-across blades and the associated holder blocks for them. These can hold either HSS blades or insert blades, although the insert blades tend to be taller and it's hard to find HSS blades that size. There are two types of the holder blocks, one is like the one I believe Willie is rumoured to have, and those slip into a regular No.1 or No.2 holder block. The problem with those is that you're moving the cutting point further out and away from the support of the compound slide. These are like the ones I sold off:
The other type of blade holders, which is what I moved to, are the integral blocks like these:
That type keeps a parting blade as close as possible to the compand slide, and seem to be more rigid. The problem is that the #7-71, #7-71-32, #771's (all makers seems to use slightly different catalog numbers) are harder or
IMPOSSIBLE to find for the smaller QCTP sets such as AXA-100 & BXA-200 sizes. Mine are CA-400, and I have seen some for the CXA-300 size QCTP's.
One other style of carbide parting insert holder is the "monoblock" style, meaning the square shank holding the insert fits directly into a No.1 or No.2 style block. I've got a couple that I use now & then. You can see the top view of one in this photo, the second one from the right.
Here's a side view of a smaller one for small diameter work:
The ones I hate the most and have had the most crashes with are the lantern tool post style, expecially those with the "flex" slot in them. I guess they put that slot in them because they 'expect" you to have the work climb up on the tool and snap off.