What is the official definition of first-hour delivery of hot water tanks?
From the energy.gov page on sizing water heaters:

To properly size a storage water heater for your home -- including a heat pump water heater with a tank -- use the water heater's first hour rating. The first hour rating is the number of gallons of hot water the heater can supply per hour (starting with a tank full of hot water). It depends on the tank capacity, source of heat (burner or element), and the size of the burner or element.
The EnergyGuide label lists the first hour rating in the top left corner as &quot;Capacity (first hour rating).&quot; The Federal Trade Commission requires an EnergyGuide label on all new conventional storage water heaters but not on heat pump water heaters. Product literature from a manufacturer may also provide the first hour rating. Look for water heater models with a first hour rating that matches within 1 or 2 gallons of your peak hour demand -- the daily peak 1-hour hot water demand for your home.

If you want the gory details (such as the outlet temperature of 125°F used in the tests), by the way, you can find them in 10 CFR 430, Subpart B, Appendix E, section 5.3.