So long as the mains-powered 18 V DC supply is floating (and transformer isolated from the mains for safety), it can replace the floating battery in the single-battery to split-power-rails versions shown in your reference.
The author does not discuss the 220 uF capacitors, but surely your mains-powered supply will have at least that much capacitance at its output (filter capacitance). Is your power supply regulated by a 7818 or similar device?
In general, op amps require capacitance between their positive and negative power pins for stability, usually something like 100 nF close to the device along with the power-supply filter capacitance.
The main thing about rail splitters is to ensure that the "gnd" output (whether a power ground or virtual ground) can supply enough current in the correct direction to handle any imbalance in current between the positive and negative outputs. If you know you have a substantial difference in those currents, you can use a output topology that only supplies current in that direction.