To answer the original question - no, hard landings
per se are not reportable
to the FAA.
From 49 CFR 830
§ 830.5 Immediate notification.
The operator of any civil aircraft, or any public aircraft not operated by the Armed Forces or an intelligence agency of the United States, or any foreign aircraft shall immediately, and by the most expeditious means available, notify the nearest National Transportation Safety Board (Board), field office1 when:
(a) An aircraft accident or any of the following listed incidents occur:
(1) Flight control system malfunction or failure;
(2) Inability of any required flight crewmember to perform normal flight duties as a result of injury or illness;
(3) Failure of structural components of a turbine engine excluding compressor and turbine blades and vanes;
(4) In-flight fire; or
(5) Aircraft collide in flight.
(6) Damage to property, other than the aircraft, estimated to exceed $25,000 for repair (including materials and labor) or fair market value in the event of total loss, whichever is less.
(7) For large multiengine aircraft (more than 12,500 pounds maximum certificated takeoff weight):
(i) In-flight failure of electrical systems which requires the sustained use of an emergency bus powered by a backup source such as a battery, auxiliary power unit, or air-driven generator to retain flight control or essential instruments;
(ii) In-flight failure of hydraulic systems that results in sustained reliance on the sole remaining hydraulic or mechanical system for movement of flight control surfaces;
(iii) Sustained loss of the power or thrust produced by two or more engines; and
(iv) An evacuation of an aircraft in which an emergency egress system is utilized.
And it probably wasn't a hard landing just an 'elevated flare'
Not that I ever did one of course.