Sigh. I think that part of the problem is the conflation of current with voltage.
In a power supply, you want the voltage to remain as stable as possible, with as little ripple as possible - with “ripple” being the fluctuation of voltage above or below the set point. The purpose of placing a capacitor between the input and ground and another between the output and ground is to try to keep the voltage ripple magnitude as small as reasonably possible.
If the current going into the supply and the current coming out of the power supply is steady, as in no fluctuations or changes in magnitude over time, then this is trivial, and you wouldn’t even need capacitors. However, in a real-world application, this is never the case.
On the input side, if you are using an AC supply, the voltage will be a sine wave at 50Hz or 60Hz, depending on where you are in the world, and it is passed through a bridge rectifier which converts it into a series of positive-going voltage “humps” at 100Hz or 120Hz (see the red line in the image below). If you place a capacitor on the output of the bridge rectifier, it smooths out the humps to a degree (see the blue line below).
Image from EETimes article:
What the capacitor is doing is absorbing charge (which is the definition of current) during the positive portions of each hump of voltage. The current changes as the voltage changes, and this is “ripple current”. The higher the capacitance, the higher the ripple current, and the smoother those voltage humps will be. The ESR of the capacitor affects how much heat will be generated when that current is drawn in or released. Lower ESR allows higher ripple current without excessive heating within the cap. The diagram below shows the internal real-world model of a capacitor. It consists of a parallel resistance (which is so large that it can usually be ignored) a series inductance (which is important only at higher frequencies) and the ESR, which is relevant at any frequency.
From Wikipedia:
Where ripple current really comes into play is on the output side. As I said earlier, if the load current is steady, there’s not much of a problem. However, for dynamic loads (as in rapidly changing current demands, such as a motor turning on and off, or a loud deep base note in an audio amp circuit), having a capacitor with high ripple current capability allows large amounts of current to flow with less heat build-up, while also maintaining the voltage level better than one with higher ESR. Notice that with the internal series resistance, as high current passes through it, not only does it dissipate power, but there will be a voltage drop across it. This causes distortion in the output voltage.
Therefore, a capacitor with higher ripple current/lower ESR, all other things being equal, should be the preferred product.