It means "dinner" only in the sense that in English, for the longest time, we referred to the midday meal as "dinner" and the evening meal as "supper." Though "dinner" has now shifted to mean the evening meal, some dictionaries still preserve this traditional definition.
From what I understand, Обед means STRICTLY "the midday meal." So if you want the traditional definition of "dinner" to be accepted, you'll have to make a case to the mods.
We would not omit "lunch" or the sentence itself would make no sense. I think your answer was rejected because of the unnecessary article "a". "do you want to have a lunch" is simply unnatural to my American English ears. It could possibly be accepted, try reporting it and explaining why you think it should be accepted. :)
While I understand that хотеть is literally "to want", обедать is a verb "to have lunch", and «___ нравиться» is "like" , I would think that "Would you like lunch?" would actually be a reasonable, better, and more natural translation of «Ты хочешь обедать?». Any native Russian speakers disagree?
So maybe my understanding of Russian grammar is bad or I'm just confused but my last example was Что вы едите. The verb to eat has that ending. Now this question has the verb's ending as хочешь. If both are used as questions directed at a singular "you" then what is the difference in the between the two endings?