Yeah, your C30 (rotary gear select) hub probably came with the
DLC34 push-push shifter and should say "C30" above the gear indication window. The quite similar
DLS30 push-push shifter for the S30 (traditional pull chain gear select) hubs says "S30" above the gear indication window.
You are correct, it does say C30. I confused myself by trying to compare shifters on Sturmey Archer's web site. I should know better than that by now, because they are notoriously slow about updating things like that.