Hi Michael,
To answer your question, the command "Write to Input Register n" writes the data to the input register while the "Update DAC Register n" updates the DAC register n from their respective input registers.
What you have thought of regarding the process is correct. I think that you just overlooked on one function of the part and that is the /LDAC function. Looking back at the part's functional block diagram, the input register comes before the DAC register. The LDAC controls when you are to load the data from the input register to the DAC register.
The command "Write to Input Register n" will write the data on the input register. Depending on the setting of your /LDAC the data will/won't be loaded to the DAC register. I'm guessing your /LDAC pin is tied to low. This would make the command "Write to Input Register n" to immediately load the data to the DAC register without having to execute the command "Update DAC Register n".
You may control the behavior of the /LDAC thru the /LDAC pin or by registers. You may want to take a look at the "/LDAC FUNCTION" sections on page 26 of the Rev.H of the data sheet for details.
I hope I have explained clearly. Please let me know if you have any further questions.
Regards,
Mark