I guess accuracy here refers to the cycle-to-cycle jitter. That is, how much the clock edge position varies when looking at single cycles.

I guess stability here refers to wandering. That is, how much the single cycle error accumulates when looking at e.g. 1 million cycles.

Accuracy measures how close the frequency is to the target, on average. Stability measures how the frequency drifts over time (and temperature, etc.). Accuracy is more of an absolute measurement while stability is more of a relative measurement. From the article:

The ticks of any atomic clock must be averaged for some period to provide the best results. One key benefit of the very high stability of the ytterbium clocks is that precise results can be achieved very quickly. For example, the current U.S. civilian time standard, the NIST-F1 cesium fountain clock, must be averaged for about 400,000 seconds (about five days) to achieve its best performance. The new ytterbium clocks achieve that same result in about one second of averaging time.

and

[U.S. civilian standard cesium reference clock] NIST-F1's performance is described in terms of accuracy, which refers to how closely the clock realizes the cesium atom's known frequency, or natural vibration rate. Accuracy is crucial for time measurements that must be traced to a primary standard. NIST scientists plan to measure the accuracy of the ytterbium clocks in the near future, and the accuracy of other high performance optical atomic clocks is under study at NIST and JILA.

So it sounds like accuracy is defined in terms of how well the clock reproduces the ideal frequency of the physical process it's based on. Hopefully there's a physicist or two around who can give us the exciting details.

Imagine a mechanical clock that has such a heavy minute hand that it goes much faster down than up. But it returns after exactly one hour, and even after ten years, it shows the full hour accurate to the second. This is a clock with very low stability, but quite high accuracy (for a mechanical clock, for an atomic clock that would of course still be terrible accuracy).

On the other hand, imagine a mechanic clock which doesn't have this stability problem, but the pendulum is not compensated for temperature changes. That is, when it gets warmer, the pendulum gets slightly longer and the clock goes slightly slower. Now temperature doesn't change very fast, so you'll not notice the effect in a short time span. However over time, the clock will drift away from the correct time, unless you manually correct it. This is clock with good stability, but not so good accuracy.

I work with atomic clocks for a living. You are on the right track.

Stability is the amount of variation over time. Clock frequency changes as parts age, temperature varies and so forth. Sometimes the variation is predictable and can be compensated for, sometimes not.

Accuracy is how close the frequency is to the specified one. A cheap ebay atomic clock can do 1 second per century.

There is also jitter which is the cycle to cycle variation of the output, which could be a square or sine wave. For some applications it matters.

What is the reference clock against which other clocks are measured?

Good question. For things like drift you can compare a new and an old clock, or compare one in a cold environment to one in a cool environment. Actually, most atomic clocks have a heater built in that maintains a stable temperature internally.

Even just comparing two clocks of the same age under the same conditions is useful, because they will never be at exactly the same frequency so will eventually drift apart.

Beyond that it's a question of determining what the uncertainties are using physics. In other words it's a mathematical proof, rather than the result of an actual measurement. Obviously measurements are made to make sure the clock is close to other known accurate clocks, but the ultimate determination of frequency accuracy is a paper exercise.