My parents were British and used traipse in 2 ways - “We had to traipse all over the place to find…” and “Take your shoes off. Don’t you dare traipse that mud all through the house.” In either case it was walking accompanied by unpleasantness.
That is an interesting idea. Somewhat recently I encountered the assertion that in English every word is sufficiently specific that there are not literal synonyms. Another interesting idea.
I chose the weary definition. What came to my mind was, declining a proposed hike saying, “I’m not interested in traipsing through the woods for hours.” But traipse is not a word I use often - if ever. I’m not sure I have any clear definition/usage in my head.
Given my frustration with a number of language usage threads lately, I’m hesitant to express much of any definitive opinion even on definitions. I’m welcoming the ascendancy of Humpty Dumpty! 
My traipsey feeling is more goal oriented than meandering, so it isn’t aimless. And it’s not necessarily pleasant or amusing, so not always whimsical. To me the key factor is semi-detachment. You might feel whimsy or you might roll your eyes, but either way you aren’t deeply affected by your journey.
I do think traipsing is more often pleasant than not.
I admire authors who can effectively use a word in an off center way to perfectly convey an action, mood, or character. It indicates mastery of the language.
Interesting.
My POV is traipsing is always resigned acceptance of excessive travel or effort. It’s never positive in any way. It might not rise to the level of vexation, but if you had a practical alternative you’d use it without a doubt.
I thought it meant to walk happily with deliberateness.