I have paint thinner as well, and mineral spirits. But I was thinking that turpentine was better to use for this. I have an idea garnered from "who knows where" that turpentine is better to thin varnishes and polyurethane, so guessed that might be true of stain as well.

Which SHOULD I use? What might happen with the wrong solvent? Thank you.

You SHOULD buy stain in the shade you like. Any of the solvents you have will thin oil based finishes and stain,turpentine is just more expensive and slow to dry compared to others.

klem1 is right. Messing around trying to lighten up an old can of stain with thinner or solvents is asking for frustration, and could produce crappy results because you would be altering the chemistry substantially. Stain is not that expensive. Buy a new can in the right shade and get on with the project.

Thanks!
The risk of altering the chemistry is not where I want to go. I am glad I got
some advice first. I’ll wait on this project til I can get out and buy a fresh
can.

To avoid the "grabbing" that you saw, wet the wood with the recommended cleanup solvent before you apply the stain, and instead of sloshing on a lot and trying to hit the right moment to wipe it all off, wipe on thin layers with a rag dipped in well-mixed stain until you like the looks.

Good tip, thanks! I was not expecting walnut to be so absorptive. But my next project is walnut too, so I want to play it safe.

Different species of wood have different density/porosity, and different boards from the same species also do depending on tree growth factors and how the wood was prepped. So if at all possible, test your stain on some scrap or on an unobtrusive part of the piece to avoid surprises.

Walnut is a wood that isn't particularly dense and is fairly porous, as you have discovered. It often is treated so that the blonder sap wood darkens to match the dark heartwood, and these two parts of the tree may be of different density.