I recently cut one of mine back. All I did was cut it on a woody area, put some growth hormone on it, and stuck it back into the pot. I did use the non-service end of a wooden spoon to poke a hole in the soil so the growth hormone would not be knocked off. -- It is looking great and has some really good new growth. So it seems happy enough to me.
I know there are other ways like layering or air layering, but I just did it the easy way.
The info for the two procedures is different. Dividing is usually only done for plantings that were started as several individual cuttings in a small pot, unless you have a larger/older plant that is producing basal (from the base or roots surrounding the base) sprouts. Simply separate the young plants (sooner is better, as far as the age of the material is concerned but the best time to do this is in Jun, where you live) and pot them up in a fast (draining) soil.
Cuttings: The best cuttings will have some wood in the stem (not all green). You can take tip cuttings & stick them in a well-aerated medium (like perlite or screened Turface) and keep them in bright light but not full sun, and keep the medium damp but not moist. No rooting aid is necessary.
You can also cut 2-3" semi-woody pieces of plants that are growing well (in good health) and just laying the cuttings atop a damp, fast (draining) medium. If there are leaves on the pieces, leave only 1 and it should stand up fairly straight. If there are no leaves, simply let them lie on the soil, not oriented in any particular way. Keep the medium damp but not wet, and tent the cuttings. Healthy pieces will form roots quickly.