i'm not sure that "signing device" & "wallet" are mutually exclusive. early in the iOS discussion they called it "skeuomorphism" right? that was supposed to be a good way to onboard users to the digital tools... for better or worse.
Skeuomorphism is a term most often used in graphical user interface design to describe interface objects that mimic their real-world counterparts in how they appear and/or how the user can interact with them. A well-known example is the recycle bin icon used for discarding files.
but as a question of branding, "Bitcoin Signing Device" is a great thing for @nvk to sell on.