[Not~Yet]

I thought it meant her "name"? I've always thought it's just like that and never given any more thought to it

I want you to stare into my eyes and say my name.
I want you to hold my hand and tell me it'll be ok.
Help me; if you don't, I don't think I can make it.
If this is a lie, then at least keep the lie going to the end.

Please understand, that's not what I'm saying.
Please notice, this isn't what I want.
Please let me go, I don't want to go there.
I've been searching for just one word.

So it doesn't literally mean her name Ayumi but more like, who she truly is and by what "name" the universe gives her, sorta like losers and winners in no more words.

EDIT: so the "word" imo isn't yet known in the song since she is still looking for it, so it's hard to say. Of course there's the theory of it going like "I am... Rainbow". I am -> She doesn't know what she really is, Rainbow -> She has come to see she's "a rainbow".