"биться" doesn't mean "to be beaten". It is either "to fight" or "to beat" (as in "the heart is beating"). The word has other meanings too, but I don't think any of those would apply here
thanks for the correction so how would this be in english? in a sentence
what does it mean by : I may be hairsplitting, but I'd say "Your grey name is rain"?
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You translated one of the lines "Your name is grey - rain". I think that "You grey name is rain" is truer to the original, even though the difference isn't very big