This is what the kids refer as "running a node", the software that defines the Bitcoin protocol consensus rules, validates transactions and has a copy of all bitcoin transaction history. Bitcoin Core is the most installed implementation. But there are alternatives such as Bitcoin Knots and Libbitcoin. This is the most important part of the whole Bitcoin stack.
Bitcoin Client →
This is quoloquialy described as "The Wallet" (due to historical reasons). This is the software most users prefer to create and check transactions. The middle-ware software that has business logic with user's personal preferences for transacting. These come in two main flavours, Light (SPV) and full-node. The full version is essentially using the "Node" localy to store/validate all the transaction data (ie Specter Desktop). The "Light" version uses someone else's node on a server (ie Electrum on desktop and Nunchuk on the phone.
Bitcoin Signer →
With the term "Signing Device" or "Bitcoin Signer" we hope to disambiguate. This 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 trully own, sign. (example are COLDCARD, Trezor and Ledger). You may have heard about these devices described as "Hardware Wallet", a misnomer. These are not Bitcoin wallets, they are signing devices. A device to store and manage private keys securely. Bitcoin wallets have chain state (access to the blockchain/ledger) and are capable of creating transactions.