Lockring tool seems to be 1mm or 2mm too small to fit sram 8 speed cassete?

I’m trying to replace a cassette. I got a chainwhip tool set + two lockring tools. I assumed it would be compatible as it’s a ‘common’ 8 speed cassette from sram.

However after trying this yesterday, the lockring seems a tiny bit too small (though all the grooves seem to line up. But it won’t budge with the nut going through the lockring. I even tried to gently hammer it in, but that just left a ‘hexagonal’ marking on the lockring…

Potentially I need to remove the nut? However there is very little contact area (a few mm) that could occur between a spanner and the nut, unless perhaps it’ll come off with very little force (I could give this a try tomorrow).

I did find this thread that seems to mention a similar problem: Is this an unsuitable cassette lockring tool?

However it’s not clear in that question on how to determine one from the other?

My main question here is:

1. What lockring tool do I need?
2. Do I need to remove the nut?

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Actually, this is starting to make sense. In the question I linked, it seems likely that they had a freewheel, and their lockring tool was too BIG. In my case I have the opposite problem, but it seems I have an actual cassette rather than a freewheel. In which case my lockring tool seems to be for a freewheel rather than a cassette..