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Paying by credit card is a multi-step process where only the initial authorization is made at the point of sale, with the finalized transactions usually entered as a batch at the end of each business day. In the United States, for service (as opposed to goods) transactions, the finalized charge is often different from the initial authorization amount, as it includes any tip that the customer may have left.

Does the restaurant's credit card system typically store the customer's credit card information after the initial authorization until the charge is finalized? If so, for how long is it typically stored? If not, how is the business able to change the amount charged to the card?

I wouldn't regard that article as knowledgeable about credit card processing systems, however:

Does the restaurant's credit card system typically store the
customer's credit card information after the initial authorization
until the charge is finalized? If so, for how long is it typically
stored?

They shouldn't do, unless their business is fully PCI-DSS security compliant.

If not, how is the business able to change the amount charged to the
card?

When the restaurant processes the card for the initial hold, they will get an identity/authorisation code back for that particular transaction. They can then submit an additional/amended transaction using that identity/authorisation code, which their card services provider can use to refer back to the initial transaction and re-use the card details as originally processed.

Merchants should ideally not be in possession of customer credit card data. Where merchants have payment terminals provided by a credit card processor (those machines you tap with your card or slide in to read the chip), the merchant doesn't even see the credit card number. Since the merchant receives a transaction identifier, they have the means to amend a transaction without having access to the CC number.

The following may now be obsolete, but one CC system I worked on allowed a mobile (no radio) POS terminal to accept credit cards for purchases. The transactions were batched up on the device until it could be docked and the data uploaded to the credit card processor (perhaps at the end of a day). PCI compliance required the transactions to be encrypted in a fairly specific manner to ensure that the stored data could only be decrypted by the processor. Given the advances in wireless communication since then, I would expect that there is no longer a need to store transactions on the device, and that they are uploaded immediately.