deionized rinses

This is a tough question to answer without more info, but here is what I suspect:

Using DI water is a good way to ensure the quality of the water is consistent (or at least as consistent as the DI filtration unit). Some sources of city water have variable quality, particularly in very seasonal locations. The city water may be good during the summer, but poor during the spring, etc.Using DI water helps ensure that you know what is (or isn't) in your water.

Also, some chemistries are incompatible with each other. Your chemical supplier should know more about the specifics based on what you are using. Also, some chemistries react unfavorably with the dissolved "goodies" in city water, so using DI water can be beneficial.

Rinse water flow rate will also play a significant part. Assuming that the incoming water is compatible and clean enough for your process, you ultimately want to minimize the amount of carry-over from one process stage to the next. Generally, the higher the flow rate of the rinse, the better, although there are other ways to achieve sufficient rinsing with low flow rates, usually by utilizing multiple rinse tanks.

DI water is almost always used in the final rinse of a process to prevent water spots when the part dries. Again, you can start with a fresh tank of DI, but if it becomes dirty with use, it is no longer "DI".

So, to answer your question: It depends. Your process will determine what kind of water you must use, be it city, RO, DI, or some other filtered water.