That site is
not as wrong as you make out. The site is clearly talking about
mains, which in terms of electric shock capability is effectively infinitely capable.
Protection trips (Residual Current Breakers; Residual Current Circuit Breakers) often trip out at 30mA. A mains circuit can easily supply say 5000mA, depending on the fuse fitted. The site is saying that 110V mains is safer than 240V mains in terms of electric shock hazard, and that is totally correct.
Voltage
IS important because low voltage can't drive enough current to cause a problem. If you short-circuit a 1.2V NiMH cell ("battery") you can probably get >2000mA. However the 1.2V cell ("battery") poses
NO SHOCK HAZARD.
Voltages have to be up above about 30V before electrical safety standards start worrying about shock. (You can get arc/flash damage though

, but that is a separate subject.)