OK, I'll put you lot out of your misery.

KE is correctly defined here as the energy required to set an object into
specific motion, assuming no losses.

Referring to my previous experiment, two masses, M and m on frictionless
wheels are connected by a compressed spring which is connected to one of them.
Initially, one observer, Os, is at rest with the pair.
When the spring is released, the two will move apart, one at speed v relative
to Os and the other at V. Another observer, Om, is arranged to be already
moving away from Os at v.
Initially:
           M||||m--------------------------------Om>v
              Os
After release of spring:
V<-M/\/\/\/\---------------------m>v-------------------------Om>v
             Os
Let the ratio of the two masses be r, such that M = rm
From conservation of momentum,
after spring release, mv = MV
V = vm/M = v/r

In the frame of P:
Initial KE = 0
Final KE = 1/2 (mv^2) + 1/2(MV^2)
         = 1/2(mv^2) + 1/2rm(v^2/r^2)
         = 1/2.mv^2[(r+1)/r]
(Since no losses are assumed, that must also be the initial PE in the spring.)

In frame Om:
Initial energy due to motion  = 1/2(M+m)v^2 = 1/2.mv^2(r+1)
Final total energy is all kinetic = 1/2M(v+V)^2
                                  = 1/2rm)(v+v/r)^2
                                  = 1/2rmv^2(1+1/r)^2
Increase in KE = 1/2mv^2[((r+1)^2/r) - (r+1)]
               = 1/2mv^2[(r^2+2r+1)-r^2-r]/r
               = 1/2.mv^2[r+1)/r]...which is the same as for frame Os.

A similar analysis for other observer speeds will produce the same equation.

This shows that whenever energy is used to set an object in motion, the total
increase in kinetic energy is the same in all frames.
Kinetic energy should not be confused with the total work that can be done
when a moving object is brought to rest, which is of course, frame dependent.
That quantity is generally not the same as the amount of energy required to
set that object into that same motion because there is always additional
energy associated with the recoil of another mass. The larger that recoiling
mass, the closer those two quantities become but irrespective of their
relative sizes, the total increase in KE is frame independent.

For example, when a car accelerates, so does the whole Earth. The total
increase in kinetic energy derived from the burning of its fuel is the sum of
the car's and the Earth's individual increases. That sum has the same value in
ALL frames (assuming no losses).

Henry Wilson DSc.