The truth is, very few people in America probably pronounce any of these words correctly, they are Japanese words being spoken by American people.
I am from the South man, you probably couldn't understand half of what I say anyway if you are from anywhere father North than Va or father West than Texas! Lord knows we don't have a clue how to say any of this Bonzay crap 'round here!
Hell we even spell stuff different here! Just ask my CrEpe Myrtles about that...
If some AMERICAN is politely trying to correct you how to say a Japanese word, you should probably politely encourage them to go... Have relations with a lawn mower.
A more serious answer though is that with such varied dialects in our country and Japan, there IS no ONE WAY to pronounce most of these words... Even the word Bonsai itself can be pronounced a handful of ways depending on where you are and who you ask (Websters dictionary lists three or four different phonetic break downs for it alone). If you get close enough to effectively relay the message of what you are trying to say, that should be good enough.
Nothing grates my nerves more than some American go talks with zero accent normally, but they have some Italian connection in their family (or maybe they are just from jersey and think that counts?) so they say "Mozzarella"as if is spelled "Mozzzzarrrrrrrrrell", adding a 30 second tongue roll to it! Or Ricotta like their tongue just unzipped itself... They think all of a sudden they got to sound like some stereotype Pizza chef from an 80s movie! Just Stop... Please. Giada- that hot chock on Food network- is the worst offender! People who are snobs about how you say these Bonsai terms are comparably annoying IMO.
Nebari, Mame, Myogi... Whatever man.. Call them roots, call it informal upright, call it SMALL. This is AMRCA DNGIT!