A wallet address is NOT a thing that identifies or points to a wallet.
that is correct. without having the unencrypted wallet file itself or the master key of HD wallets, there is no way of linking different addresses together claiming they belong to the same wallet.
a wallet creates a signature/hash FROM a public/private key pair that is contained within the wallet. This signature is used to authenticate a transaction without actually exposing the key pair.
a wallet creates the signature from the private key (not the pair) and in a transaction you publish the signature + the public key, and both of these are used to "unlock" the transaction output that is being spent, or in simple terms spend bitcoin.
A wallet therefore is a container for one or more private/public key pairs and the algorithms required to generate signatures from them.
Did I get that right?
yes this last one is the best explanation of what a wallet is. although wallet could simply refer to the file or the medium used to store the key(s) like paper wallet or wallet.dat file that only contain the key(s).