The exercise date triggers your tax treatment, even though the shares are not delivered until the settlement date. When you exercise the option, a transfer of property occurs, and you acquire a "beneficial interest" in the stock under IRS Regulations (see the IRS regulations on Internal Revenue Code Section 83 discussed in the Tax Court case Walter v. Commissioner
The tax treatment of the income at exercise depends on whether your stock options are NQSOs or ISOs. (The taxation of stock appreciation rights is similar to that of NQSOs.) When you sell all the shares at exercise (i.e. in a cashless exercise or a same-day sale), the income from the spread at exercise is ordinary income for NQSOs, ISOs, and SARs.
This website has detailed content about tax withholding and reporting in the Tax Center, which includes annotated examples.