How to refer to the upper, flat part of a cliff?

A cliff is basically divided in two parts: the "flat" surface where you stand and then fall. What do you call the former? Here's an example of the desired usage:

There were two climbers on the cliff: one on the safety of the [...], and the other paralyzed, hanging from a thousand-foot fall.

I think you are looking for clifftop.

An area of land at the top of a cliff: the windswept clifftops [OD]

As opposed to cliffside:

the steep side of a cliff or of any abrupt natural incline of considerable size [MW]

Enjoying the high life! Thrill-seekers living on the edge as they jump on clifftop 2,300ft above Norwegian fjord ... with no safety rail

The Trolltunga - or Troll's Tongue in English - sits 700 metres above the Ringedalsvatnet Lake in Southern Norway

[dailymail.co.uk]

I googled this:

The two parts of a cliff referred to are the horizontal part or the pla·teau /plaˈtō/ noun: plateau; plural noun: plateaux; plural noun: plateaus 1. an area of relatively level high ground. synonyms: upland, tableland, plain, mesa, highland, coteau "a windswept plateau"

And, the vertical part is referred to as the cliff-face (klɪf feɪs) or just, the face noun the vertical face of a cliff. "People have been banned from climbing the cliff face because it is too dangerous."

A 'clifftop' just refers to any top of a cliff. A 'plateau' is any flat elevated geologic surface. An 'overhang' is a portion of a structure or formation that protrudes from the main body and rests such that it is 'overhanging' the ground (hanging above it). When used in the context of overhanging cliffs, the term 'overhang' refers to the part of the cliff that actually has no direct downward connection to the ground. I think the word most accurately describing the portion of the type of cliff you are talking about is 'overhang', though if the cliff is strictly vertical 'clifftop' may have to suffice.

While clifftop is accurate, it is a word that climbers rarely use. Or at least all the climbers I've ever met in the United States.

Any flat part of the cliff is a "ledge" and the last flat part of the cliff is the "summit ledge". Generally, the summit ledge is where you unrope and the climb is considered to be over. In your example above, a climber would use "summit ledge".

However, that might not convey the exact meaning to a general reader. It depends whether you want to use the jargon of the sport to convey expertise, or to be more widely accessible. Also a summit ledge isn't always the top of the cliff, but everything above that is considered easy and safe enough to travel without a rope.