Nice to read your reply
Lynn, glad it was a useful post.
I am not too familiar with key clicks being used in flute music (I’m still stuck at Bach et al.)
but I got out my alto flute and open hole (standard soprano) flute and started tapping some of the keys/holes out of curiosity, not blowing or giving any air at all.
I noticed if you hold down the thumb left hand and press another key for the click, like LH middle finger it will make a different sound than if you do not have the LH thumb down due to changes of resonance of the bore/where the sound escapes from.
And with different combinations of keys down (similar to a recorder’s forked fingering) while key clicking another key(s) I could almost hear differences for each combination, some were quite different from others.
I do not have my closed hole soprano flute with me at the moment so could not compare open vs closed holes and if that makes any difference (I presume it does) and I heard the same kind of sound differences on my alto flute with the pitch of the clicks being lower.
It probably depends on how you want your key clicks to sound/pitch, as to which holes/fingers are down at the time, and which are being pressed for the clicks.
I found a reference to blocking the embouchure hole also, causing the pitches to lower:
Recorders
I got out my alto and tenor recorders, I found the same kind of result, a bit quieter (both wood), again, depending on which holes were closed, and tapping an open hole somewhere, resulted in different sounds. The tenor was more resonant than the alto obviously.
Then I got out my bass recorder’s (Aulos plastic) middle joint, placed it on some soft plastic horizontally on the table and “played” the holes a bit like a piano, holding down a “chord” (combination of holes, sometimes forked) then tapping an open hole/key, different combinations. I stopped before the fun ran out 
Just out of interest, are you coming to all of this from playing a Paetzold (or similar) recorder?
(or in the part, add a QR code for a video of what is required and how it is to sound? hmmmm…!)
Berio, yes that triggered some memories, never attempted to play any of his music, well beyond me! Thanks for mentioning him.