How do cesium atomic clocks measure time? I am trying to understand cesium atomic clocks better.
I am not getting HOW the cesium oscillation is actually being counted. 
So from my understanding of an older atomic clock: 


*

*Cesium gets heated.

*Heated cesium goes through magnets.

*Excited cesium atoms get discarded and non-excited cesium atoms get hit with microwave radiation at about $9,192,631,770 \, \mathrm{Hz} .$

*This excites electrons to the hyperfine splitting upper energy level of the $6\text{s}$ orbital.

*Electrons that get excited hit a detector, and the ratio between these and the total number entering the microwave chamber is calculated, subsequently the frequency of the microwave generator is adjusted to reach the highest ratio. To this point to me all we did was measure that cesium has a two energy levels separated by some energy,$$
{E}_{\text{photon}}
~=~ h f
~=~ h \times {9,192,631,770} \, \mathrm{Hz}
\,,$$where $h$ is Planck's constant and $f$ is a photon's frequency.
So great – but how is this actually being counted? Is there a device somewhere that says – ah ok, maximum cesium atoms are being excited, let's start counting... start counting what though?!?  And what device is doing the actual counting, as in: $1,$ $2,$ $3,$ $4,$ ..., $9,192,631,770,$ and then saying, BOOM, that's a second.
 A: Nowadays it's reasonable to make an actual digital counter that operates at 10 GHz. Even so, you might rather use a prescaler to divide this frequency down to 1/4 or 1/8 of the microwave frequency before you actually start counting cycles.
In olden times you might have generated your 10 GHz signal by multiplying up a lower frequency (maybe in the 100's of MHz?) and you'd be tuning that low frequency to adjust the microwave frequency. Then you would count the cycles of the lower frequency to measure off a desired time interval. (Note I don't know how early atomic clocks were designed, this is just one practical way of doing it)
Edit:
The July, 1968 Hewlett-Packard Journal (on-line here) gives this block diagram for how a rubidium frequency standard was built at that time:

So my guess wasn't too far off.
As far as I know, commercial atomic clocks typically give a 10 MHz clock as output, and they never expose their 9.19 GHz clock (or 6.83 GHz in the case of rubidium) to the user.
A: The counting is done by the signal generator.
The clock contains a signal generator that generates the 9,192,631,770 Hz signal going to the microwave transmitter. This generator contains a counter that counts every cycle the generator creates (though as The Photon says it may use a prescaler).
