Thank you for reaching out to me.
I believe the twelve words are the passphrase. Is the mnemonic phrase something other than the passphrase? I haven't used any Bitcoin Core wallet until now, at least not successfully. If I have another wallet, where might I find it? I'm happy to generate a new wallet but how will I then be able to send my coins to the new one without being able to authenticate the sending request with the passphrase that I either don't know or am inputting incorrectly?
What passphrase exactly do you mean? the password or the 12 words seed?
Bitcoin core wallet doesn't have seed phrase feature if you use bitcoin core wallet it only generates a wallet.dat file. as your backup.
If it's asking for a password it is not a 12 words seed you should put the password that you use when you created the wallet before.
If you can't remember the password(passphrase) the only option that I know to open your wallet is by brute-forcing.
Check this guide from here to brute-force your bitcoin core wallet
https://github.com/gurnec/btcrecover
Okay, it looks like I may have another wallet. The one that now possesses funds is Bitcoin Core - Wallet with the orange/yellow B emblem. The other is Bitcoin Core - Wallet [testnet] with a green B emblem. Could this be my original wallet? It's presently not encrypted, but I can choose to encrypt it from the settings menu, whereas with the other wallet it is encrypted and I can choose to change the passphrase but only if I know the existing passphrase, which is the crux of the whole problem to start with. What steps do you recommend I take please?
I don't exactly understand what you are pointing out.
If your funds are from Bitcoin core testnet and you are trying to send it to Bitcoin core Mainnet they are different testnet coins are no value.
Check the wallet which is no testnet sign. If it is asking for a passphrase like I said if you can't remember the password the only solution for this is by using a brute-force tool the link already posted above.