OK, I think I’ve got it figured out. I just went out and looked at my antenna (in a light rain, had to get some other stuff out of the car). On top of that tube you see the top of the antenna, a 1/4 inch or so tube, then the threads. You are pulling on the antenna and the rim that you see, but the main sleeve that you should be pulling on / gently prying up (I believe) is the one whose rim is snug against the threads. It looks like a thread, but you’ll see that it really an inner sleeve that is snug against the threads. The antenna is inside of that sleeve, then something connects it at the bottom of that sleeve (I didn’t go that far in looking at my antenna just now).

As I remember, this is one of the steps that also confused me when I changed my antenna, though somehow I luckily saw that other sleeve after ten or fifteen minutes of playing around with it. Again, it's hard to see, it looks like a thread, but you'll notice that it's really a rim. Gently pry up on that, see if that is the problem.