Specifically, does it refer to SHA hashes per second, or does it refer to nonce-iterations per second, because to check a nonce requires the application of SHA twice.
The number reported is an estimation of the number of block headers iterated through by all miners on the network.
This means it refers to double-SHA256 hashes performed, not broken down to individual SHA256 operations. This makes the most sense, because even in the very early days (2010, perhaps before), no full SHA256 instances were being computed. Several optimizations let you skip parts of the computation.
It is an approximation, because the actual hash rate is not observable. It can be computed by looking at the created blocks on the network, and treating each as (2^256 / target) worth of attempts (so approximately 4.3 billion hashes per difficulty per block). That is the expected number of necessary hashes.
By looking at the blocks, another inaccuracy is introduced: only blocks that end up in the main chain are counted. Some blocks end up on temporary forks because another miner produces another nearly simultaneous competing block that wins. This is only a fraction of a percent today, but it does waste a small amount of hashrate, which also ends up not being counted.
Hash Rate The hash rate is the measuring unit of the processing power of the Bitcoin network. The Bitcoin network must make intensive mathematical operations for security purposes. When the network reached a hash rate of 10 Th/s, it meant it could make 10 trillion calculations per second. Source