And I also bought from the same website, a 6.35mm collet here
I thought these would fit together, but they don't.
Is there any clue to what I did wrong in the selection, or how I can fit the 6.35mm collet I got to a proper clamping nut...?
Is your Krees electrospindle in the list of the suitable units for both collet and nut? If yes, you probably received a defective product or a wrong one.
Why not switching to ER collet and collet nuts, they are cheaper and easier to find on the market.