What do the security definitions of SHA-3 mean, quantitatively?
Aleph's comment and pointer to the Keccak reference goes a very long way to answering this question (p. 9):



The key words here are "workload equivalent to $N$ calls to $\mathrm{Keccak-}f[r + c]$ or its inverse."  Invocations of the Keccak permutation are being used as the unit of time.  Your intuitive take that it's hash function invocations isn't very far off from this, but note that a single SHA-3 call may result in one or more permutation calls, so Claim 1 is more stringent.

In a more abstract setting, where we're talking about the sponge construction in general and not a concrete implementation like Keccak, we talk about the permutation (and its inverse) as an oracle that the adversary can freely call (often people say "query") as they see fit.  Then the security of the construction is expressed in terms of how many queries the adversary makes.  The oracle calls are treated as outputting the result of a random permutation in constant time.

See also: the Keccak team's more generic treatment of sponge functions.