Are the additional confirmations a count of the subsequent blocks built atop the block that provided the original confirmation?
Yes.
Would it be fair to say that a transaction's first confirmation, the mining of the block that includes it, is a slightly different type of confirmation than the subsequent 5?
Not really. The number of confirmation simply indicates the number of blocks that would have to re-mined in order to reverse a transaction (since each block includes a hash of the previous block, altering an early block invalidates all later blocks). The only practical difference is that for the first confirmation, miners can choose whether or not to include your transaction in the next block (they may refuse it if the fee is insufficient, for example), but once it's in a block (it has one confirmation), it will be additionally confirmed by all subsequent blocks (since miners can't simply decide to remove transactions from a previous block). This is why a transaction will sometimes take an extremely long time to get the first confirmation, then gain additional confirmations at the usual rate.
