How Does the FAA Define the Dimensions of the Touchdown Zone? Does the FAA give a regulatory definition of the term &quot;Touchdown Zone&quot; for use within the context of landing operations?

In 14 CFR 91.175 the FAA requires—in certain cases—that pilots must be in a position to land within the &quot;Touchdown Zone&quot;:

(c) Operation below DA/DH or MDA. Except as provided in §91.176 of this chapter, where a DA/DH or MDA is applicable, no pilot may operate an aircraft, except a military aircraft of the United States, below the authorized MDA or continue an approach below the authorized DA/DH unless—
(1) The aircraft is continuously in a position from which a descent to a landing on the intended runway can be made at a normal rate of descent using normal maneuvers, and for operations conducted under part 121 or part 135 unless that descent rate will allow touchdown to occur within the touchdown zone of the runway of intended landing;

Is this zone explicitly defined? If so, where is the applicable definition found?

No definition is offered in 14 CFR 1.1. However, several uses or definitions of this term do exist in various FAA advisory documents. These uses or definitions are generally similar, but include variances. A few of these include:
AC 91-79A (referencing an obsolete document):

the TDZ is referred to as a point 500-3,000 ft beyond the runway threshold not to exceed the first one-third of the runway

AIM 2–1–5:

3,000 ft beyond the landing threshold or to the midpoint of the runway, whichever is less.

The Pilot Controller Glossary:

The first 3,000 ft of the runway beginning at the threshold.

Also of note, the runway touchdown zone markings on long precision runways generally extend to 3,000 ft beyond the threshold.
Found this definition on an FAA website today:

The runway touchdown zone is usually defined as 1000 feet from the runway threshold or 1/3 the total available landing distance. This provides runway “underrun” in case the pilot comes up short of his/her aim point, as well as increased obstacle clearance while on final approach.