"Signing Device" is a better term because it frames the process in terms of speech/action.

"This is my signature",
"using my signing device".

It also help show how non-signing wallets (ie "hosted" or "web" wallets) take away your capacity to truly own, sign.

https://twitter.com/nvk/status/1518756527649271809

https://twitter.com/nvk/status/1519028112465580036 [Nitter]

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Bitcoin Signing Device - Because 'wallet' is too confusing
#23035
https://signingdevice.com