Edit: Actually I don't think the picture for the Hyperfine states below is quite correct. Below I describe it as if the two clock states have different spin projection ($m_F$) along the $z$ axis, but actually the two Cs clock states are mainly different by their total spin, $F$. I'll try to update this with a more correct picture later.
In a microwave clock the ground and excited states are two different hyperfine states of the atom. Intuitively these states can be pictured as follows. In both the ground and excited the electron is spin up and has angular momentum in the $+z$ direction. Imagine the electron orbiting the nucleus in the $x-y$ plane. The nucleus also has a spin and associated magnetic moment. In the ground state the nucleus is pointing down and in the excited state the nucleus is pointing up. These two states are split in energy because the angular momentum of the electron (plus the charge of the electron) creates a magnetic field at the location of the nucleus. Thus, there is a (small) energy difference depending on if the nuclear magnetic moment is aligned or anti-aligned with the magnetic field of the electron.
Your description of the Cs clock is mostly correct. I will add some more detail about how the actual spectroscopy is performed to measure the "absorption dip". Initially the Cs is prepared in the ground state. Then a pulse of light is applied to put the Cs in a superposition of ground and excited states. Recall that a superposition of spin up (along $+z$) and down (along $-z$) is the same as a spin pointing in the $x-y$ plane. Exactly which way the spin is pointing in the $x-y$ plane can be controlled by controlling the polarization and phase of the magnetic field vector of the incident microwave light.
Suppose the experimentalists choose the spin to be pointing along the $+x$ direction after the first pulse. Now the nuclear spin begins to precess about the magnetic field created by the electron (which is still present) at a rate given by the energy difference between the two levels. If one waits for the spin to precess many many times ($N$ times) and then applies a pulse again when the nuclear spin is again pointing exactly in the $+x$ direction then the atom will be brought exactly to the excited state after the end of the second pulse.
After the second pulse we can measure whether the atom is in the ground or excited state. If the microwave pulses were exactly the right frequency of microwave radiation then the atom will always be in the excited state. But if the pulses were the wrong frequency the atom will be in a superposition of ground and excited states after the second pulse so sometimes they will be in the ground state sometimes they will be in the excited state. The sensitivity of this measurement increases as the time allowed for spin precession is increased. This is how atomic physicists are able to measure the Cs precession (or energy splitting) very very precisely. What I have described is a Ramsey sequence You are correct that once the measurement is performed feedback is enabled to lock the frequency of the microwave signal (or local oscillator) to the appropriate frequency. In analogy to a regular clock: the atoms are like the quartz oscillator or the pendulum, they provide the frequency reference, the microwave signal (which is locked to the atomic rotation) is like the hands on the clock.
Your second question is what is the analogy for an optical clock. Well above I have described the microscopic nature of the two states used for a microwave clock. They are two hyperfine states of the atom which are mostly different because of the direction the nuclear spin is pointing. To come up with the analogy for an optical clock we need to figure out the microscopic picture of the two states in an optical clock.
In the optical clock we are considering two atomic states separated by an optical transition. In the ground state the electron is in an $S$ orbital and in the excited state the electron is an a $P$ orbital. This means in the ground state the electron is in a little sphere around the nucleus and in the excited state the electron is a dual lobe ($P$ orbital) around the nucleus with the lobes pointing in the $+z$ and $-z$ directions.
If the atom is put into a superposition of the $S$ and $P$ state (now using a pulse of optical rather than microwave light) then one can imagine the electron charge distribution (or wavefunction) to slosh back and forth from the $+z$ side of the nucleus to the $-z$ side of the nucleus. See this nice post and animation https://physics.stackexchange.com/a/293413/128186.
The same Ramsey sequence is now performed. But now instead of measuring the frequency of precession of the nuclear magnetic moment we now measure the frequency of the sloshing of the electrons spatial distribution. One question is what causes the sloshing? Well, it is exactly the coulomb potential felt by the electron as a result of the charged nucleus. If the electron is too far above the nucleus it feels a force pulling it down and then like a harmonic oscillator it overshoots and ends up on the $-z$ side where it is then pulled upwards.
This is my first time hearing about a nuclear clock. I looked it up briefly and based on the above I'll give my guess as to how this would work. In the proposal I found for a nuclear clock the idea is for the two clock states to be two different isomers of the nucleus. This means the nucleus will have the same protons and neutrons but they will be in a different physical configuration. If there is some way to drive a transition between these two states (perhaps with a laser?) then it is possible to drive the system in a superposition of the two states. Once that is done the nucleus will oscillate between those two states as above. We can again perform Ramsey spectroscopy to measure this frequency of oscillation. I guess that in this case it is sort of similar to the optical case in that what will be oscillating will be the relative positions of the protons and neutrons. The interesting thing here is that instead of the restoring force being mediated by the electromagnetic field (coulomb force between electron and nucleus) it will now be mediated by the strong force which binds the nucleus together. So here the nucleus is to be thought of as a blob of matter which is sort of quivering at a high frequency because of the strong force. This seems like a really nice probe into some fundamental physics of the strong force if you could perform such an experiment!