There’s a distinction between verification and confirmation.
Verification is checking that some data, such as transaction data, meets all the Bitcoin rules. For example you could check that the sum of outputs of a transaction does not exceed the sum of its inputs. You could check that none of the inputs have already been spent. You can check signatures. You can check that the locking script is well-formed. There are many such checks. Verifying a transaction does not tell you whether the payment has been made (received).
Confirmation of a transaction is nothing more than a count of the number of blocks that have been added to the blockchain which are either the block that contains that transaction or which are linked back to that block. 0 blocks means unconfirmed, 6 or more is regarded as irrevocably confirmed. Full nodes can count this number of blocks themselves. So when you make a payment, confirmation of your transaction is what you are interested in. Confirmation is what tells you the payment has happened, that it has been irrevocably received.
A transaction-ID is simply a representation of a number that uniquely identifies some specific transaction. This Transaction-ID can be used by a node to index a list of transactions to aid retrieval. For example some software might create an index of Transaction IDs that points to the block that contain each transaction.
An unspent transaction output (UTXO) is effectively an unspent coin. A node can and should check that the inputs to a transaction of interest are all UTXOs. This is part of how Bitcoin prevents double-spending. Each full node typically keeps a list of all UTXOs and removes spent ones and adds newly created ones as it processes each new transaction.
Simplified Payment Verification is the section of Nakamoto’s white paper that contains the text quoted in the question.
Full nodes do all the verification themselves. Lightweight nodes, such as those that use Simplified Payment Verification (SPV) rely to some extent on other nodes to perform verification. Unlike full nodes, they partially trust some other nodes to be honest. So the quoted text does not apply to full nodes.