Clamping
I'm not sure we have addressed the issue of clamping. This circuit is used (well, was used in pre-digital days) to establish an exact DC level for AC signals. The prime example in my experience was TV circuits: the varying modulation (AC) needed to be superimposed on a fixed bias Voltage (DC).
Now, as to how it works, you are nearly right in your thoughts about the diode, but remember the Voltage across the capacitor is in series with the signal source. When the capacitor is charged to the peak Voltage of the AC signal, then the diode is never forward biased at any part of the signal. For example:
If sig = +1V and cap = 0V, then diode is reverse biased, so 1V across Resistor.
If sig = -1V and cap =0V , then diode is forward biased, it conducts, no V across R.
But this has charged up the capacitor to 1V, so
now sig = +1V and cap = +1V, so +2V across diode, again reverse biased, so +2V across Resistor
now sig = -1V and cap = +1V, so 0V across diode and resistor.
If now sig = 0V and cap= +1V, then +1V across diode (reverse bias) and +1V across R.
So input varies from +1 to -1 and output varies from 0 to +2.
The varying part of the signal (AC) remains the same, but now the average Voltage is +1V, rather than 0V.
Depending on the resistance of the source and the diode during conduction, it could take several cycles for the capacitor to reach full charge.
Depending on the load resistor, some of the charge on the capacitor must "leak" through the load, so that it needs partial recharge each cycle.
In practice we design with a high load resistance and low source resistance to make these effects negligible.
To allow for the non-zero forward Voltage drop of the diode and to allow precise adjustment, we can take the non-signal end of the diode to a DC voltage provided by a pair of resistors (or potentiometer.)
In case it might help, I've done an Excel simulation of your circuit and plotted graphs of the voltages. These are attached.
Clamp1 shows reasonable values. Clamp slow has bad values, so that the capacitor charges slowly and discharges too much through the load resistor.
The Voltage plots have a consistent scale, but the current plots are arbitrarily scaled to fit.
How ever you understand capacitors, it is reasonable to think of them as blocking DC and as passing AC.
You can see from the graphs, the exponential charge of the capacitor from 0 to -1V, this being the DC voltage developed by the diode from the 1V p-p AC signal.
The Voltage across the load resistor can easily be seen as the AC signal added to this DC Voltage on the capacitor.
( If one actually solves the differential equations for circuits like this, the solutions come out having two parts, one exponential function and one sine function.)