Power adapter giving a specified current
As you say, the current depends on the attached resistance. 

The rated current is not what they will provide, but what they can provide. If your powersupply is rated 5 V, 1 A, it means that it can deliver up to 1 A of current. 

So if I take that supply and connect to it a 5 kOhm resistor, the supply will provide only 1 mA of current. If I connect a 50 Ohm resistor, it will provide 100 mA of current. 5 Ohms will be the maximum, at which point it will provide 1 A of current. When I attach a lower value resistor, all bets are off regarding what will happen. IT could be the powersupply just fails to provide 5 V at the current draw, and will for example drop to 4 V if we overload it a bit. It could also be that the supply provides the current, but damages itself because it is getting too hot internally to handle this. 

Quick added note: Sometimes the manufacturere will actually mention that the supply can operate outside of this limit, but at reduced performance elsewhere. For example, I have a few lab power supplies that are rated to 24 V, but will happily output up to 30 V. The manufacturer (HP, now keysight) says that doing so will not risk any damage to the device, but they cannot guarantee the noise and output resistance of the supply is within the advertised specifications.