The reason you can't just power a 5 volt regulator with a microcontroller is because relays are inductive devices.
I know you know what you meant, and I know I know what you meant, but this should most likely be:
The reason you can't just power a 5 volt relay with a microcontroller is because relays are inductive devices.
The next few sentences routinely refer to relays as regulators.
I'd also say statements like "..is because relays are induction devices." and "Inductive loads like motors and relays draw more current than a microprocessor can safely supply." are not entirely correct. Relays and motors are both inductive loads, but that doesn't mean they have to require large amounts of current, and high-current devices are not exclusively inductive. A better statement might use something along the lines of "Motors and relays shouldn't be connected directly to a microcontroller because they are inductive and require more current that a microcontroller can safely supply".
Even this statement isn't 100% correct though,there are certainly many low-current relays with built in fly-back diodes that can be connected directly to a uC's output. In fact, your OJE-SH-105DM has a nominal current of 40mA (from it's datasheet); the Atmega168 at the heart of the Arduino can supply 40mA with a 1V internal drop (page 320 of the Atmega168 datasheet), so, assuming you clamped the relay with a diode, you could indeed connect your relay directly to the Arduino.