First, let’s clear up some terminology. Touchdown zones are only on runways with precision approaches. Aiming Point Markings are at 1,000' in USA (ICAO standard is different and is runway length dependent) and are also on non-precsion and visual runways. If you fly a small GA airplane, you will often land in the first 1,000' using the Aiming Point as a guide. The markings are at a fixed distance (500') from each other so that’s probably why you have heard that term. I have most of the FAA publications and ACs on my computer and the only place that "fixed distance markers" is referenced is in the Chart Supplement for some runways that have non-standard distances for the markers, so I am curious as to where you have read that term. EDIT: Per the edited question, I changed my search to “fixed distance marker“ and found a half dozen references. All of them are in Knowledge Test questions and none in other FAA publications.
AIM 2−3−3. Runway Markings
e. Runway Touchdown Zone Markers. The touchdown zone markings identify
the touchdown zone for landing operations and are coded to provide
distance information in 500 feet (150m) increments. These markings
consist of groups of one, two, and three rectangular bars
symmetrically arranged in pairs about the runway centerline, as shown
in FIG 2−3−1, Precision Instrument Runway Markings.
TOUCHDOWN ZONE− The first 3,000 feet of the runway beginning at the
threshold. The area is used for determination of Touchdown Zone
Elevation in the development of straight-in landing minimums for