Imagine a water pipe with a diaphragm placed in the middle. Water pressure from (say) the left hand side distorts the diaphragm to the right - water on the right hand side of the diaphragm gets displaced and flows. There is no direct connection, both sides are completely isolated from each other, water flows due to pressure on the diaphragm. When the pressure from the left stays constant, the diaphragm reaches a stable state and no further displacement takes place. Increase the pressure slightly and the diaphragm again distorts and pushes more water out the pipe. The water pipe can now said to be "charged".
Lower the pressure, the diaphragm returns to its normal state, causing a drop of pressure on the right hand side. Water then flows back into the pipe in the opposite direction - a negative flow if you like.
In this way, a capacitor "conducts" an AC signal and blocks a DC signal. pulsating DC is DC, unless the polarity changes, it is DC - whether it is positive DC or negative DC. It is NOT AC, and there is no AC on a signal that does not change polarity - just changes of pressure, which the capacitor reacts to and allows to pass. This is why you WILL get a current flow from a DC source through a capacitor whilst it is charging. Once charged, current flow stops.
This can be proven - wire up a 555 timer to give a square wave DC signal - a capacitor will pass that signal.
Take a capacitor and charge it with a positive DC source - then send a positive pulse down the wire - the capacitor will pass that pulse, even though it is DC. The output of the capacitor is AC, that is true, the polarity alternates because of the changes in pressure causing current to flow back into the capacitor as described above.