To my mind if you say "next week" then that refers to the very next week that comes along after this week. You wouldn't mean the first week in the next month (unless it WAS also the first week in the next month). So if you say "next X-day" the next should refer not to the week but to the very next day of that name that comes along. Where is the word or concept "week" in the phrase "next Tuesday"? The next applies to the day, not the week. It means next Tuesday, not next week on Tuesday.

No matter what you say I will continue to stubbornly insist on using the most logical meaning.
(and yes I meant to split that infinitive)