Are you referring to / Do you mean?
How do you say

(1) "Are you referring to A or B", and
(2) "Do you mean A..." in Korean?

My attempts (but I'm not too sure):

For instance, "Are you referring to the slogan written in Chinese, or the original English slogan?"

너를 방금 중국어 스러건을 말 했어요? 아니면 영어 스러건을 말 했어요?

Is there are more elegant (or shorter) way of phrasing this (assuming this is correct)?

And the other question - for instance, "Do you mean that you're sick because of the food you ate earlier, or you ate because you were sick?"

무슨 말 이에요? 너를 앞서 나쁜 건 먹기 때문에 지금 아픈 거예요? 아니면, 너를 아프기 때문에 밥을 먹었어요?

감사합니다!

Question

Are you referring to / Do you mean?
How do you say

(1) "Are you referring to A or B", and
(2) "Do you mean A..." in Korean?

My attempts (but I'm not too sure):

For instance, "Are you referring to the slogan written in Chinese, or the original English slogan?"

너를 방금 중국어 스러건을 말 했어요? 아니면 영어 스러건을 말 했어요?

Is there are more elegant (or shorter) way of phrasing this (assuming this is correct)?

And the other question - for instance, "Do you mean that you're sick because of the food you ate earlier, or you ate because you were sick?"

무슨 말 이에요? 너를 앞서 나쁜 건 먹기 때문에 지금 아픈 거예요? 아니면, 너를 아프기 때문에 밥을 먹었어요?

감사합니다!

Answer

방금 중국어 슬러건을 말한 거에요? 아니면 영어 슬러건을 말한 거에요?

It's elegant enough XD.

You can use '뜻' ,'의미' also  but,  '말' is more easier to say.

중국어 슬러건을 뜻하는 건가요, 영어 슬러건을 뜻하는 거에요?
중국어 슬러건을 의미하는 건가요, 영어 슬러건을 의미하는 거에요?

But sometimes the opponent doesn't get it or confused what I mean when I say '그거 말하는 거에요? ' so sometimes I use '뜻,의미' to clarify it. '그거를 의미한 거에요?'

Or you can say shortly "중국어 슬러건이요? 영어 슬러건이요?"
but sometimes the opponent doesn't get it. :p

(지금) (무슨 말이에요? ) 전에 먹은 것 때문에 지금 아픈(/아프다는) 거에요? 아니면 아퍼서 밥을 먹었다는 거에요?
We can put "지금" to express what I don't understand or I'm doubt that you really mean without '무슨 말이에요'

지금 무슨 말이에요?
지금 말한 게 그 뜻이에요? 
지금, 당신이 잘했다는 거에요?

I hope this is helpful for you :)