So the 15-20mA is the current required to switch the actual relay ie current drawn by the coil. this is being switch be the transistor on the relay board, so all we are doing it switching this transistor on using the opto isolator so it should only require a few mA to switch that transistor, but as we don't have a diagram with component details we are only guessing at the required voltage/current.
If you supply details of the relay you are now going to use then we have a better chance of helping you .
So I just tested an opto isolator driving a transistor to switch a relay using this circuit
and it works just fine, this is essentially how the relay board should work with an opto isolator , the small current from the opto isolator transistor driving the base of the of the next transistor which switches the relay coil. So there must be something different in the way the circuit is connected on the relay board pi9 is using.