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I had a bit of a play in the shed.
Hornady shellholder on Lyman priming tool, noticeably larger hole, and plunger reaches flush at best. Shellholder 0.385 inches thick (bottom to where face of case...

Lyman definitely work, I bought a Hornady set recently and I think they were a little bit too thick (as you have found), I'll wander out to the shed and have a look.
I just keep my Lyman shellholder...

The Lyman hand tool is fussy about what shellholders you use, e.g. Lee shellholders are way too thick to work.