“depois de amanhã” apparently means ANY day after tomorrow including the day after or any other day after. It could literally mean “we should do lunch some day that will never happen”. English actually can express those differences using different words but PT seems to be somewhat vague beyond tomorrow. Is that correct?
If you look at it literally, “depois de amanhã” might sound vague, but it’s not. We only use it for the day immediately after tomorrow, i.e. two days from now. We have other expressions for when we don’t want to commit to a specific date, such as “um dia destes” 
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Muito obrigado! This clarification was very helpful. I am a bit literal myself, so I think it will be better for me to say, for example “a quinta feira, etc.” when acknowledging a meeting proposed by natives as “depois de amanhã”. I am still getting used to things like “assisting TV”, which befuddles my poor brain. I deal with these oddities by hearing the meaning as taught but avoid speaking certain idioms myself.
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