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In 1971 the Hafele-Keating experiment was conducted. It is a well-known
and well documented experiment with two planes, carrying two atomic
clocks. One clock flew in western direction around the earth, and one
flew in eastern direction around the earth. The purpose was to prove
"time delation" as predicted by Einstein's theories of Relativity.

The results of the experiments were presented as a big success, proving
the theory of Special Relativity. The atomic clocks onboard the planes
did walk at a different rate than a stationary atomic clock that was
used for reference.

In reality the results were hardly conclusive.

However, the results did not PROVE Einsteins Special Relativity but
DISPROVED it. One clock walked indeed slower than the reference clock,
but the other clock walked FASTER than the reference clock. This
phenomenon was explained by noting that one of the clocks flew counter
to the rotation of earth, and so was moving even slower than the
reference clock.

Hm... peculiar! I thought that central to SR was the fact that each
inertial frame is equivalent, and there is no "zero" inertial frame AKA
"aether". However, the Hafele-Keating experiment showed that the faster
one moves with regard to an inertial frame against which also earth
moves (through its rotation), the slower your clock goes.

Then ask your school money back. ;-)
SRT is based on inertial frames, just like classical mechanics. The rotating
earth surface is not a valid inertial frame for such a global experiment.
Similarly for GPS the earth centered inertial frame is used.

So there is a "zero" inertial frame (aether), and experiments conducted
in this "zero" frame have results that differ from experiments
conducted in inertial frames that move with regard to the "zero" frame.
The main difference is that in the zero-frame, clocks walk fastest.

Wrong (but an often seen mistake). Any inertial frame can be chosen, but
there are preferred frames for simplest calculations - again just as in
classical mechanics.

The experiment showed that there IS NO RELATIVITY. There might be time
delation, but it is a phenomenon that is based on movement with regard
to an ABSOLUTE zero-frame.

That interpretation is perfectly compatible with "relativity" theory.
According to SRT, no such zero frame can be determined. That's all there is
to it; just ignore bunkum.

Cheers,
Harald