Heck, if i knew that i'd be a shihan myself. But looking at various defs
and a narrow def of ayumi
as a "crossing step", maybe i understand more how the terminology i've been taught is different.
My shihan describes "ayumi" variously a tactic (a natural-looking walking movement so as not to give one's intentions away) and a technique (to use the momentum of the entire body to power strikes and takedowns). So instead of "ayumi" being one technique and "irimi" another, all techniques are defined as extensions of "ayumi".
On some level, it is just a difference of naming conventions, but i also think it is supposed to be psychological. Perhaps the intent is to resist labeling and therefore distinguishing different techniques (so much of what I am taught is just called "this"), which i believe is done in other arts (particularly Chinese) as well. It may be a way of training reflexive responses instead of "kamae, enter, block, counter, takedown, stop, rest". But guess what i am still doing.
Don't ask me to explain much more. Totally at my limit of understanding here.