Its from the song "Stay with me";
Also, is there a better way to translate the 'naide' form of a verb than "without doing"? It sounds quite odd I think.
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This could be the quoting
"Don't go!", [she] cried.
Or the the contidional one (though it's unlikely here , as mentioned by @snailplane below):
When [she] said "Don't go", [I] cried.
To tell which it is, we'd need more context.
I found the lyrics (「真夜中のドア～Stay with me」 by 唐沢美帆) and the song goes like this:
私は私 貴方は貴方と 昨夜言ってた そんな気もするわ グレイのジャケットに見覚えがある コーヒーのしみ 相変わらずなのね ショーウィンドウに二人映れば Stay with me... 真夜中のドアをたたき 帰らないでと泣いた あの季節が今 目の前 Stay with me 口癖を言いながら 二人の瞬間を抱いて まだ忘れず 大事にしていた
So it seems the first version is correct, except the subject is the singer.
I cried, pleading "Don't go!".
(changed slightly to avoid the double meaning of the English verb "cry")
The other stanzas also use the quoting
と many times.
～ないで, here it is short for
～ないで下さい, i.e. it's a request to not do something (go back in this case). See also Why don't we use ～なくて instead of ～ないで?