Whether it's better or not for your purposes is of course your call, but as I said to chaosmosis above, I resolve this tension in my own mind by understanding "koan" as you use it to mean "exercise."

Then again, I also replace all of your Japanese phrases in my head with their corresponding English.

I suspect this just reflects my not valuing a particular kind of myth-building very much in this context, so I just experience it as a mildly annoying distraction.
If you find it valuable, by all means continue with it.