Are all 'metal wallets' functionally equivalent? Within the context of wallets, is it correct that a metal bitcoin wallet needs to store both public and private keys?
I'm looking at some metal wallets and want to clarify that the're all basically the same in that they'll give you different bits and bobs which are then must be physically arranged?
Or, are some metal wallets etched or otherwise &quot;written&quot; and then sent to the user?  Because that seems a very trusting act, to provide a private key to some third party.
Just dealing with bitcoin.  Perhaps naively thinking of storing public/private key &quot;written&quot; down with metal.
I would think something like:

minting physical bitcoins

would the point of such a wallet.  Obviously not minted, but just storing the public/private key on metal for reference.
Just to be clear/nitpick, the products you've linked to are designed to act as highly resilient physical backups for a bitcoin wallet. It doesn't really make much sense to refer to these metal plates/etc as 'metal wallets', as they cannot perform any normal wallet functions (send, receive, etc). Rather, they are just a way to store a backup for the seed that will (re)create your wallet using the appropriate software.

Within the context of wallets, is it correct that a metal bitcoin wallet needs to store both public and private keys?

Not explicitly. These days when you create a new wallet you will likely be presented with a mnemonic seed phrase, in the form of 12/24/etc words. These words are the backup for your wallet, if you input them into some wallet software (and follow the correct derivation path) then you will get access to your wallet.
So these metal backup plates are built to have your seed phrase engraved into them in some way or another. By using metal, you can protect your backup against things like fire or water damage.
Note that you could engrave a private key itself into metal, but given the chance of transcription error, you can perhaps see the reason that mnemonic seed phrases are very popular: they are human-readable and thus easier to error-check. A mistake could be extremely costly! You don't want to lose access to your bitcoin.

I'm looking at some metal wallets and want to clarify that the're all basically the same in that they'll give you different bits and bobs which are then must be physically arranged?

The function of them is the same: keep your mnemonic seed phrase safe. So if you use one, you should choose a design that you find to be:

durable
easy to set up
easy to understand
secure against accidental scrambling (what happens if you accidentally drop it? etc)
easy to store securely

Maybe there are some other considerations, but those are the obvious things that come to mind at least.

One consideration that is less obvious: purchasing a product like that can be really bad for your operational security / privacy. The company you purchase from will likely have a database of all past customer information. As example: not long ago, there was news that a major hardware wallet manufacturer' customer database was hacked, and customer information was stolen (note that no BTC was compromised). Since then there have been reports of phishing attacks targeting those people, since they are obviously bitcoin users.
With that in mind, consider the option of instead just purchasing:

an alphabetical metal punch set
a hammer
a stack of wide flange washers
a nut/bolt of appropriate size to fit the washers

You can then stamp each word onto a washer. Be sure to number them as well, so that you can ensure they don't get scrambled out of order (this would be very bad!). If you want, you can just stamp the first 3-4 letters of each word, since no two words in the BIP39 list have the same first 3 letters.
This is a very simple, secure, and private way of backing up your wallet, which will probably be cheaper than any of the products listed on that site, especially after considering the price of shipping.
This is what a metal punch set looks like, for reference:


Or, are some metal wallets etched or otherwise &quot;written&quot; and then sent to the user? Because that seems a very trusting act, to provide a private key to some third party.

NO. You should absolutely not ever use a pre-engraved product, creating private keys is something that the user should not be trusting a third party to do. You leave yourself at risk of theft from the third party, any one of their contractors, postal workers, customs agents, etc.

I would think something like: 'minting physical bitcoins' would the point of such a wallet. Obviously not minted, but just storing the public/private key on metal for reference.

The Casascius coins you linked were created as collectibles, more than secure offline storage, but in principle the function is similar. Note that those coins contain the info for private keys, that were generated by the coin's creator, so he is a trusted third party in this case (in spite of the warnings above).
You may also be interested in a product called the OpenDime (disclaimer: I have nothing to do with OpenDime, and have never used one, but they look cool), it allows for physical transfer of bitcoin without requiring trust that your counterparty will not keep a copy of the private keys. Note however, that the Opendime is NOT recommended for longterm physical storage, so while it is physical, it is functionally different than the backup methods described above.