"Homo" is Greek for "same" or "one" or "identical"; "ousios" for "nature" or "being" or "substance" or "essence". Is the Son identical with the Father, possessed of the same substance as the Father? Or is the Son merely similar to the Father, only like Him? (And if only like the Father, how like: a little bit like or a lot like? And if even a lot like, is a "miss" here "as good as a mile"?)
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***So I say, let those guys call themselves "homooousios-arians", and let Trinitarian be defined by the Word of God.***
[1] Arius rejected the term homoousios and used the term homoiousios. You have a typo here, I think.
[2] Word of God for LDS = the Bible plus LDS "Scriptures" plus whatever might be revealed next week minus what was taught by a LDS prophet that was later viewed as speculation [i.e. Adam-God nonsense].
[3] Heavenly Mother, pre-mortal spirit babies, Jesus and Lucifer as brothers, celestial plural marriage, dunking for the dead so they get a second chance to become Mormons... It's a wild ride.
Anthanasius used the term homoousios (same substance/being) while Arius used homoiousios (similar substance/being), as you said. They had a big debate in Nicea over that "iota of difference", and the homoousios guys won.