My supervisor and I just made such a misunderstanding today. He interpreted “next Tuesday” as the (1) nearest coming Tuesday while I interpreted it as (2) the Tuesday of the next week. His native language is Dutch, and my native language is Chinese. I also asked another person whose native language is Dutch and he agreed to the first interpretation. But in Chinese, I would say definitely everyone agrees to the second interpretation. When I did a rough search through the English forums today, my conclusion is that most English users also agree with this second interpretation, which is described as the old-fashioned usage by the OED.
So from a language/culture point of view, maybe Dutch speakers prefer (1). The Dutch word for "next" is "volgende", which probably is closer to the meaning of "the following; which comes directly after", so they prefer (1) (I cannot be sure of this as I'm not learned in Dutch). In Chinese, the literal translation of "next Tuesday" is "next-week's-day-4" (where people usually count from Monday as 1), so we would combine the "next" with "week" together. Thus, we prefer (2), day of next week.
Back to English grammar. I would say English is for sure more similar to Dutch than to Chinese. But still, English users tend to favor interpretation (2), day of next week, as well. My explanation is that, "next" is a restrictive modifier in "next Tuesday". Since there is only one Tuesday in any one calendar week, we prefer this restrictive modifier explanation, and the word "next" actually applies to modify the upper level of "Tuesday", that is, it is the week that shall be the next, and we draw the Tuesday from this week. So this should also extend to "next June" = "June of next year". This practice makes English more efficient, since "next Tuesday" uses fewer letters than "Tuesday of next week"; and for the Tuesday of this week, the efficient counterpart is still defined, "this Tuesday". (Actually another tricky thing is, we omit the definite article "the" before "next" -- I'm not sure whether this plays a role in the meaning here.)
However, if the modified word is not the only one in the cycle/upper level, we tend to think that "next" modifies this word itself. Eg, "the next student" does not mean the student from the next classroom, "next chapter" doesn't mean the chapter from the next book.