Yep, I've discovered the wonders of E-Tube Project! It told me two interesting pieces of information: 1) It confirmed that our info display is the SC-MT800, and 2) it informed that in fact we don't have a separate wireless unit installed. That clears up the mystery about how come it is that our bike seems to work with two wireless units installed...it doesn't!
I also learned something else I hadn't known: that there is a another set of switches built in to the brake levers that I had not known existed. But the best part is that I was able to re-configure the set of switches to behave in what is (for me) a rational way. The default behavior just made no sense to me at all.
For those who are new to this (as I am), there are some valuable tidbits of info that may help you. One is that you can use the battery charger to connect your bike to your PC and you can run E-Tube Project from there.
The other thing I've learned is that Shimano's documentation sucks eggs, at least as far as downloading, installing, and using E-Tube goes. In hindsight, it all makes sense, but if it's your first time through...
Terry, thank you very much for helping me through this. Life is much better now.