Thanks for the additional suggestions and questions. I think you're onto something asking about my OS. But first...
* I first tried plugging my Teensy directly into the Mac (I had been using a hub),using a different USB cable and got the same behavior.
* I do have other MIDI devices (Keystation 88 Keyboard, LPD8 Controller) and they all keep working fine when Logic is running.
* When I'm in Logic, and look at Preferences \ MIDI \ Inputs, I see "Teensy MIDI" listed and it is bolded and checked, so Logic sees my Teensy.
* I don't know if this is important, but when I connect my Teensy the OS gives me a message saying "Changed Number of MIDI Inputs/Outputs. 5 Inputs, 3 Outputs" and when I unplug it, I get the same message but with "4 Inputs, 2 Outputs". So my Teensy adds both 1 Input and 1 Output, but it only needs to act as in Input to my Mac.
But...I am still on Sierra (10.12.6) and running Logic version 10.4.3, both of which are getting a bit old. The reason is that (with the exception of this new issue with my Teensy controller), my system has been absolutely rock solid; no Logic crashes, no annoying bugs, everything just works perfectly. I'm due to replace my Mac, and would do so right now, but I want to wait until the new MINI's with Apple Silicon can handle more RAM, so I've been keeping my current system going and not wanting to mess up a good thing.
One reason that I suspect my slightly outdated operating system and/or Logic version could be the reason for this strange behavior is that a while back a friend gave me his 2014 Mac MINI to use when I was traveling and that system also has Sierra (10.12.6) and Logic (10.4.4). I just tried my Teensy controller on that system and it exhibited the exact same behavior. I plugged in my Teensy, loaded MIDI Monitor and could see that the small portable MIDI keyboard worked fine before and after Logic was loaded, but my Teensy controller worked fine before Logic started but stopped as soon as I loaded Logic. I'm not sure why this would happen, but it certainly points toward the OS or Logic version as the culprit.
So I think I have three options:
1) Not use my Teensy controller until I upgrade to a new Mac MINI with Apple Silicon and more RAM (hopefully in the next year?)
2) Upgrade my Mac's operating system and Logic (perhaps to Mojave - but I'm quite hesitant to do this!)
3) Try loading my Teensy with software that makes it look like an MCU (https://github.com/tttapa/Control-Surface#readme
) and see if that makes any difference.
Thanks again for the very helpful suggestions. Please let me know if you disagree with my conclusions above, or if you have other options I could consider to resolve this issue with my current OS & Logic.