I use either either and neither neither nor neither either.

So you say both then?

I don’t say either.

In my head I pronounce it as “either”, but out loud I pronounce it like “either”

I don’t. I say either.

Probably eye-ther, but eee-ther on random occasion

Yes.

Either either or either, depending on maybe the context or the phonetics of words used. Kinda like how you say the or the depending on if the next word starts with a vowel or not.

Either works, really.

Growing up, I self-taught myself to pronounce this word as “Ee-ther”. Then I went to a private teacher and learned that the correct English pronunciation is “Eye-ther”. Then I also came to know that the word can be pronounced in both way regarding different accents. Then I stopped caring which pronunciation I was using while speaking. Most of the time, I say “Eye-ther”. I utter the later one barely.

Data or data?

Neither

Tomato tomato.

Whichever one you say, I say the other one.