What is the FAA's definition of "for hire"?
Without combing through the regulations, and the vast collection of LOI on the subject, I will give a general answer.

For hire for the purposes of 91.409 applies to the carriage of persons. So the carriage of a student pilot, with a flight instructor is "for hire."  The carriage of people receiving a scenic tour that they paid for, is "for hire." 

If you belong to a flying club, essentially operated as a partnership, then the aircraft is not for hire, as the fixed and variable expenses are shared among the members of the club.  Essentially, you are paying your share for your aircraft, assuming the club or partnership owns the aircraft.  In that case the landing light and 100 hour examples do not apply.

In general, if operating part 91, and the pilot is paid in furtherance of the part 91 entity, then for hire does not apply. This would include for example a corporate pilot, paid by the owning entity of the plane.

In general, flight instruction is not considered a for hire operation, when the person receiving instruction is one of the "owners" of the plane, as in a partnership or flying club.

So, you have some examples for your second to last question.

For the last question, it covers the operation, and the purpose of the operation. So your paid pilot flying your TBM taking your family to your private island, is not for hire.

There are no less than 20 FAA Letters of Interpretation which have been written on this topic. The interpretations have even impacted states and how they charge sales and use tax. 

Addendum #1: For the FAA, under 91.409, the term "for hire" applies to the carriage of the aircraft.  So if a flight school owns a plane, and a student gets instruction in the aircraft, it is subject to 100 hour inspections, because the student is being carried by the plane and CFI which the flight school is providing.