For electric water heaters that can be obtained with various different options for the size and power draw of their heating elements, will larger elements use more power overall? For example, let's say an electric water heater can be configured with elements that draw 3 kW or 4.5 kW, with the 4.5 kW model delivering a greater first-hour yield. But will the 4.5 kW model use more power in the aggregate? Or will it use around the same overall amount of power as the 3 kW model, but simply deliver its heat faster so as to lower its runtime?
You'll use more power with the larger element, unless you/your family are very strange. You'll use more power because you'll get more hot water before the water heater can't keep up - if you used exactly the same amount/temperature of water, you'd use exactly the same amount of power. Standby losses are pretty minimal for a well-insulated modern hot water heater, and standby losses would be almost the same - except for when they are (very) slightly higher as the tank recovers faster so it's not as cool.
You may need to upgrade wiring if the element size is an increase.
If you want to save money or energy on heating hot water, a heat pump [either separate from the water heater and connected to it (might outlast the water heater), or a "heat pump water heater" (presumably needing expensively thrown out when it dies)] or a desuperheater on your A/C (if you have and use A/C) or a solar thermal hot water system (I seem to recall you are in a hot desert area where this should work well) are all far more effective than fiddling with element size. Naturally they all have capital costs and added complexity.