I agree, but I like to think it means the first one. Honestly, I don't know that "don't tread on me" works in that vein perfectly either, just because it's not as simple to understand. Maybe "mind your own business?"
As a NW Ohioan, i like "Don't Give Up The Ship" since it carries a similar ideal without all the baggage of "Don't Tread On Me" or "An Appeal To Heaven".